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Thus, we write
$$q(V,T) = \sum_{\substack{j \text{(states)}}} e^{-\varepsilon_j/k_B T}$$
(17.53)
We will call sets of states that have the same energy, *levels.* We can write *q(V, T)* as a summation over levels by including the degeneracy, g., of the level: <sup>J</sup>
$$q(V,T) = \sum_{\substack{j \text{ (level... | {
"Header 1": "**13–3.** Vibration–Rotation Interaction Accounts for the Unequal Spacing of the Lines in the *P* and *R* Branches of a Vibration-Rotation Spectrum",
"Header 2": "**17–8.** A Molecular Partition Function Can Be Decomposed into Partition Functions for Each Degree of Freedom",
"token_count": 374,
"... |
- **17-1.** How would you describe an ensemble whose systems are one-liter containers of water at 2SOC?
- **17-2.** Show that Equation 17.8 is equivalent to *f(x* + *y)* = *f(x)f(y).* In this problem, we will prove that f (x) ex *eax.* First, take the logarithm of the above equation to obtain
$$\ln f(x+y) = \ln f(x) ... | {
"Header 1": "**13–3.** Vibration–Rotation Interaction Accounts for the Unequal Spacing of the Lines in the *P* and *R* Branches of a Vibration-Rotation Spectrum",
"Header 2": "**Problems**",
"token_count": 2044,
"source_pdf": "datasets/websources/biochem/F814BC5915875384820.pdf"
} |
17-20. Deriving the partition function for an Einstein crystal is not difficult (see Example 17-3). Each of the N atoms of the crystal is assumed to vibrate independently about its
lattice position, so that the crystal is pictured as N independent harmonic oscillators, each vibrating in three directions. The partit... | {
"Header 1": "**13–3.** Vibration–Rotation Interaction Accounts for the Unequal Spacing of the Lines in the *P* and *R* Branches of a Vibration-Rotation Spectrum",
"Header 2": "**Problems**",
"token_count": 1990,
"source_pdf": "datasets/websources/biochem/F814BC5915875384820.pdf"
} |
Show that the fraction of harmonic oscillators in the ground vibrational state is given by
$$f_0 = 1 - e^{-h\nu/k_{\rm B}T}$$
Calculate $f_0$ for $N_2(g)$ at 300 K, 600 K, and 1000 K (see Table 5.1).
**17-37.** Use Equation 17.55 to show that the fraction of rigid rotators in the *J*th rotational level is giv... | {
"Header 1": "**13–3.** Vibration–Rotation Interaction Accounts for the Unequal Spacing of the Lines in the *P* and *R* Branches of a Vibration-Rotation Spectrum",
"Header 2": "**Problems**",
"token_count": 1185,
"source_pdf": "datasets/websources/biochem/F814BC5915875384820.pdf"
} |
Frequently, we need to investigate the behavior of an equation for small values (or perhaps large values) of one of the variables in the equation. For example, we might want to know the low-frequency behavior of the Planck distribution law for blackbody radiation (Equation 1.2):
$$\rho_{\nu}(T)d\nu = \frac{8\pi h}{c^... | {
"Header 1": "**13–3.** Vibration–Rotation Interaction Accounts for the Unequal Spacing of the Lines in the *P* and *R* Branches of a Vibration-Rotation Spectrum",
"Header 2": "SERIES AND LIMITS",
"token_count": 1991,
"source_pdf": "datasets/websources/biochem/F814BC5915875384820.pdf"
} |
If we apply Equation I.9 to $f(x) = e^x$ , we find that
$$\left(\frac{d^n e^x}{dx^n}\right)_{x=0} = 1$$
SO
$$e^x = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \cdots$$
Some other important Maclaurin series, which can be obtained from a straightforward application of Equation I.9 (Problem I–7) are
$$\sin x = x ... | {
"Header 1": "**13–3.** Vibration–Rotation Interaction Accounts for the Unequal Spacing of the Lines in the *P* and *R* Branches of a Vibration-Rotation Spectrum",
"Header 2": "SERIES AND LIMITS",
"token_count": 1573,
"source_pdf": "datasets/websources/biochem/F814BC5915875384820.pdf"
} |
- **1-1.** Calculate the percentage difference between $e^x$ and 1+x for x=0.0050, 0.0100, 0.0150,..., 0.1000.
- **1-2.** Calculate the percentage difference between ln(1+x) and x for x = 0.0050, 0.0100, 0.0150, ..., 0.1000.
- **1-3.** Write out the expansion of $(1+x)^{1/2}$ through the quadratic term.
- I-4. Eval... | {
"Header 1": "**13–3.** Vibration–Rotation Interaction Accounts for the Unequal Spacing of the Lines in the *P* and *R* Branches of a Vibration-Rotation Spectrum",
"Header 2": "SERIES AND LIMITS",
"Header 3": "**Problems**",
"token_count": 1577,
"source_pdf": "datasets/websources/biochem/F814BC5915875384820.... |
The energy of an atom in an ideal monatomic gas can be written as the sum of its translational energy and its electronic energy
$$\varepsilon_{\rm atomic} = \varepsilon_{\rm trans} + \varepsilon_{\rm elec}$$
so the atomic partition function can be written as
$$q(V,T) = q_{\text{trans}}(V,T)q_{\text{elec}}(T)$$
(1... | {
"Header 1": "**13–3.** Vibration–Rotation Interaction Accounts for the Unequal Spacing of the Lines in the *P* and *R* Branches of a Vibration-Rotation Spectrum",
"Header 2": "**18–1.** The Translational Partition Function of an Atom in a Monatomic Ideal Gas is $(2\\pi mk_{\\rm B}T/h^2)^{3/2}V$",
"token_count":... |
In this section, we will investigate the electronic contributions to q(V, T). It is more convenient to write the electronic partition function as a sum over levels rather than a sum over states (Section 17-8), so we write
$$q_{\text{elec}} = \sum_{i} g_{ei} e^{-\beta \varepsilon_{ei}}$$
(18.8)
where $g_{ei}$ is t... | {
"Header 1": "**13–3.** Vibration–Rotation Interaction Accounts for the Unequal Spacing of the Lines in the *P* and *R* Branches of a Vibration-Rotation Spectrum",
"Header 2": "**18-2.** Most Atoms Are in the Ground Electronic State at Room Temperature",
"token_count": 1996,
"source_pdf": "datasets/websources/... |
#### EXAMPLE 18-1
Using the data in Table 18.1, calculate the fraction of fluorine atoms in the first excited state at 300 K, 1000 K, and 2000 K.
SOLUTION: Using the second line of Equation 18.10 with $g_{e1}=4,\ g_{e2}=2,$ and $g_{e3}=6,$ we have
$$f_2 = \frac{2e^{-\beta \varepsilon_{e2}}}{4 + 2e^{-\beta \... | {
"Header 1": "**13–3.** Vibration–Rotation Interaction Accounts for the Unequal Spacing of the Lines in the *P* and *R* Branches of a Vibration-Rotation Spectrum",
"Header 2": "**18-2.** Most Atoms Are in the Ground Electronic State at Room Temperature",
"token_count": 1464,
"source_pdf": "datasets/websources/... |
When treating diatomic or polyatomic molecules, we use the rigid rotator-harmonic oscillator approximation (Section 13–2). In this case, we can write the total energy of the molecule as a sum of its translational, rotational, vibrational, and electronic energies:
$$\varepsilon = \varepsilon_{\text{trans}} + \varepsil... | {
"Header 1": "**13–3.** Vibration–Rotation Interaction Accounts for the Unequal Spacing of the Lines in the *P* and *R* Branches of a Vibration-Rotation Spectrum",
"Header 2": "**18–3.** The Energy of a Diatomic Molecule Can Be Approximated as a Sum of Separate Terms",
"token_count": 1081,
"source_pdf": "datas... |
| M<br>l<br>l<br>o<br>e<br>c<br>u<br>e | l<br>i<br>E<br>t<br>e<br>c<br>r<br>o<br>n<br>c<br>t<br>t<br>s<br>a<br>e | i<br>8<br>/<br>K<br>v | 8<br>/<br>K<br>ro<br>t | 1<br>D<br>/<br>k<br>1<br>l<br>m<br>o<br>·<br>-<br>0 | 1<br>f<br>k<br>l<br>D<br>J<br>m<br>o<br>-<br>- |
|--------------------------------------|-----------... | {
"Header 1": "**13–3.** Vibration–Rotation Interaction Accounts for the Unequal Spacing of the Lines in the *P* and *R* Branches of a Vibration-Rotation Spectrum",
"Header 2": "**18–3.** The Energy of a Diatomic Molecule Can Be Approximated as a Sum of Separate Terms",
"token_count": 1113,
"source_pdf": "datas... |
In this section, we will evaluate the vibrational part of the partition function of a diatomic molecule under the harmonic-oscillator approximation. If we measure the vibrational energy levels relative to the bottom of the internuclear potential well, the energies are given by (Section 5–4)
$$\varepsilon_v = \left(v ... | {
"Header 1": "**13–3.** Vibration–Rotation Interaction Accounts for the Unequal Spacing of the Lines in the *P* and *R* Branches of a Vibration-Rotation Spectrum",
"Header 2": "**18–4.** Most Molecules Are in the Ground Vibrational State at Room Temperature",
"token_count": 1908,
"source_pdf": "datasets/websou... |
This quantity is given by $\sum_{v=1}^{\infty} f_v$ but because $\sum_{v=0}^{\infty} f_v = 1$ , we can write
$$f_{v>0} = \sum_{v=1}^{\infty} f_v = 1 - f_0 = 1 - (1 - e^{-\Theta_{\text{vib}}/T})$$
or simply
$$f_{v>0} = e^{-\Theta_{vib}/T} = e^{-\beta h v}$$
(18.29)
**FIGURE 18.4... | {
"Header 1": "**13–3.** Vibration–Rotation Interaction Accounts for the Unequal Spacing of the Lines in the *P* and *R* Branches of a Vibration-Rotation Spectrum",
"Header 2": "**18–4.** Most Molecules Are in the Ground Vibrational State at Room Temperature",
"token_count": 740,
"source_pdf": "datasets/websour... |
The energy levels of a rigid rotator are given by (Section 5-8)
$$\varepsilon_J = \frac{\hbar^2 J(J+1)}{2I}$$
$J = 0, 1, 2, \dots$ (18.30a)
where I is the moment of inertia of the rotator. Each energy level has a degeneracy of
$$g_J = 2J + 1 (18.30b)$$
Using Equations 18.30a and 18.30b, we can write the rotati... | {
"Header 1": "**13–3.** Vibration–Rotation Interaction Accounts for the Unequal Spacing of the Lines in the *P* and *R* Branches of a Vibration-Rotation Spectrum",
"Header 2": "**18-5.** Most Molecules Are in Excited Rotational States at Ordinary Temperatures",
"token_count": 1918,
"source_pdf": "datasets/webs... |
Although it is not apparent from our derivation of $q_{\rm rot}(T)$ , Equations 18.33 and 18.34 apply only to heteronuclear diatomic molecules. The underlying reason is that the wave function of a homonuclear diatomic molecule must possess a certain symmetry with respect to the interchange of the two identical nuclei ... | {
"Header 1": "**13–3.** Vibration–Rotation Interaction Accounts for the Unequal Spacing of the Lines in the *P* and *R* Branches of a Vibration-Rotation Spectrum",
"Header 2": "**18-5.** Most Molecules Are in Excited Rotational States at Ordinary Temperatures",
"Header 3": "18–6. Rotational Partition Functions C... |
The discussion in Section 18-3 for diatomic molecules applies equally well to polyatomic molecules, and so
$$Q(N, V, T) = \frac{[q(V, T)]^N}{N!}$$
As before, the number of translational energy states alone is sufficient to guarantee that the number of energy states available to any molecule is much greater than the... | {
"Header 1": "**13–3.** Vibration–Rotation Interaction Accounts for the Unequal Spacing of the Lines in the *P* and *R* Branches of a Vibration-Rotation Spectrum",
"Header 2": "**18-5.** Most Molecules Are in Excited Rotational States at Ordinary Temperatures",
"Header 3": "**18-7.** The Vibrational Partition Fu... |
For 8vib.j = 954 K (the doubly degerate bending mode),
$$\frac{\overline{C}_{V,j}}{R} = \left(\frac{954}{400}\right)^2 \frac{e^{-954/400}}{(1 - e^{-954/400})^2} = 0.635$$
For 8vib.j = 1890 K (the asymmetric stretch),
$$\frac{\overline{C}_{V,j}}{R} = \left(\frac{1890}{400}\right)^2 \frac{e^{-1890/400}}{(1 - e^{-18... | {
"Header 1": "**13–3.** Vibration–Rotation Interaction Accounts for the Unequal Spacing of the Lines in the *P* and *R* Branches of a Vibration-Rotation Spectrum",
"Header 2": "**18-5.** Most Molecules Are in Excited Rotational States at Ordinary Temperatures",
"Header 3": "**18-7.** The Vibrational Partition Fu... |
In this section, we will discuss the rotational partition functions of polyatomic molecules. Let's consider a linear polyatomic molecule first. In the rigid-rotator approximation, the energies and degeneracies of a linear polyatomic molecule are the same as for a diatomic molecule, $\varepsilon_J = J(J+1)h^2/8\pi^2I$ ... | {
"Header 1": "**13–3.** Vibration–Rotation Interaction Accounts for the Unequal Spacing of the Lines in the *P* and *R* Branches of a Vibration-Rotation Spectrum",
"Header 2": "**18–8.** The Form of the Rotational Partition Function of a Polyatomic Molecule Depends Upon the Shape of the Molecule",
"token_count":... |
We can now use the results of Sections 18–7 and 18–8 to construct q(V,T) for polyatomic molecules. For an ideal gas of linear polyatomic molecules, q(V,T) is the product of Equations 18.43, 18.44, 18.46, and 18.50:
$$q(V,T) = \left(\frac{2\pi M k_{\rm B} T}{h^2}\right)^{3/2} V \cdot \frac{T}{\sigma \Theta_{\rm rot}} ... | {
"Header 1": "**13–3.** Vibration–Rotation Interaction Accounts for the Unequal Spacing of the Lines in the *P* and *R* Branches of a Vibration-Rotation Spectrum",
"Header 2": "**18–9.** Calculated Molar Heat Capacities Are in Very Good Agreement with Experimental Data",
"token_count": 1517,
"source_pdf": "dat... |
| · | | | Vibrational<br>Contribution | | Total $\overline{C}_V/R$ | Total $\overline{C}_{v}/R$ |
|----------|---------------------|------------|-----------------------------|-----------------------------------|--------------------------|--------... | {
"Header 1": "**13–3.** Vibration–Rotation Interaction Accounts for the Unequal Spacing of the Lines in the *P* and *R* Branches of a Vibration-Rotation Spectrum",
"Header 2": "**18–9.** Calculated Molar Heat Capacities Are in Very Good Agreement with Experimental Data",
"token_count": 728,
"source_pdf": "data... |
- **18-1.** Equation 18.7 shows that (stran,) <sup>=</sup>~k <sup>T</sup>in three dimensions, and Problem 18-3 shows that (stran,) = ~k <sup>T</sup>in one dimension and ~k <sup>T</sup>in two dimensions. Show that typical values oftranslational quantum numbers at room temperature are 0(10<sup>9</sup> ) form = 10-26 kg, ... | {
"Header 1": "**13–3.** Vibration–Rotation Interaction Accounts for the Unequal Spacing of the Lines in the *P* and *R* Branches of a Vibration-Rotation Spectrum",
"Header 2": "**18–9.** Calculated Molar Heat Capacities Are in Very Good Agreement with Experimental Data",
"Header 3": "**Problems**",
"token_coun... |
The experimental heat capacity of CO(g) can be fit to the empirical formula
$$\overline{C}_V(T)/R = 2.192 + (9.240 \times 10^{-4} \text{ K}^{-1})T - (1.41 \times 10^{-7} \text{ K}^{-2})T^2$$
over the temperature range 300 K < T < 1500 K. Plot $\overline{C}_V(T)/R$ versus T over this range using Equation 18.41, an... | {
"Header 1": "**13–3.** Vibration–Rotation Interaction Accounts for the Unequal Spacing of the Lines in the *P* and *R* Branches of a Vibration-Rotation Spectrum",
"Header 2": "**18–9.** Calculated Molar Heat Capacities Are in Very Good Agreement with Experimental Data",
"Header 3": "**Problems**",
"token_coun... |
**18-32.** The experimental heat capacity of $CH_{4}(g)$ can be fit to the empirical formula
$$\overline{C}_V(T)/R = 1.099 + (7.27 \times 10^{-3} \text{ K}^{-1})T + (1.34 \times 10^{-7} \text{ K}^{-2})T^2 - (8.67 \times 10^{-10} \text{ K}^{-3})T^3$$
over the temperature range 300 K < T < 1500 K. Plot $\overlin... | {
"Header 1": "**13–3.** Vibration–Rotation Interaction Accounts for the Unequal Spacing of the Lines in the *P* and *R* Branches of a Vibration-Rotation Spectrum",
"Header 2": "**18–9.** Calculated Molar Heat Capacities Are in Very Good Agreement with Experimental Data",
"Header 3": "**Problems**",
"token_coun... |
In general, this is very difficult, but for large *R* we can proceed as follows. We treat R, or£, as a continuous variable and ask for the number of lattice points between£ and s + !:ls. To calculate this quantity, it is convenient to first calculate the number of lattice points consistent with an energy :::; *s.* For ... | {
"Header 1": "**13–3.** Vibration–Rotation Interaction Accounts for the Unequal Spacing of the Lines in the *P* and *R* Branches of a Vibration-Rotation Spectrum",
"Header 2": "**18–9.** Calculated Molar Heat Capacities Are in Very Good Agreement with Experimental Data",
"Header 3": "**Problems**",
"token_coun... |
Thermodynamics is the study of the various properties and, particularly, the relations between the various properties of systems in equilibrium. It is primarily an experimental science that was developed in the 1800s and still is of great practical value in many fields, such as chemistry, biology, geology, physics, and... | {
"Header 1": "**13–3.** Vibration–Rotation Interaction Accounts for the Unequal Spacing of the Lines in the *P* and *R* Branches of a Vibration-Rotation Spectrum",
"Header 2": "**The First Law of Thermodynamics**",
"token_count": 728,
"source_pdf": "datasets/websources/biochem/F814BC5915875384820.pdf"
} |
The concepts of work and heat play important roles in thermodynamics. Both work and heat refer to the manner in which energy is transferred between some system of interest and its surroundings. By *system* we mean that part of the world we are investigating and by *surroundings* we mean everything else. We define *heat... | {
"Header 1": "**13–3.** Vibration–Rotation Interaction Accounts for the Unequal Spacing of the Lines in the *P* and *R* Branches of a Vibration-Rotation Spectrum",
"Header 2": "**The First Law of Thermodynamics**",
"Header 3": "**19-1.** A Common Type of Work Is Pressure-Volume Work",
"token_count": 2018,
"s... |
Work and heat have a property that makes them quite different from energy. To appreciate this difference, we must first discuss what we mean by the state of a system. We say that a system is in a definite state when all the variables needed to describe the system completely are defined. For example, the state of one mo... | {
"Header 1": "**13–3.** Vibration–Rotation Interaction Accounts for the Unequal Spacing of the Lines in the *P* and *R* Branches of a Vibration-Rotation Spectrum",
"Header 2": "**19–2.** Work and Heat Are Not State Functions, but Energy Is a State Function",
"token_count": 1918,
"source_pdf": "datasets/websour... |
SOLUTION: The expression for the reversible work is
$$w_{\rm rev} = -\int_1^2 P dV$$
where
$$P = \frac{nRT}{V - nb} - \frac{an^2}{V^2}$$
We substitute this expression for P into $w_{rev}$ to obtain
$$\begin{split} w_{\text{rev}} &= -nRT \int_{1}^{2} \frac{dV}{V - nb} - an^{2} \int_{1}^{2} \frac{dV}{V^{2}}... | {
"Header 1": "**13–3.** Vibration–Rotation Interaction Accounts for the Unequal Spacing of the Lines in the *P* and *R* Branches of a Vibration-Rotation Spectrum",
"Header 2": "**19–2.** Work and Heat Are Not State Functions, but Energy Is a State Function",
"token_count": 216,
"source_pdf": "datasets/websourc... |
Because the work involved in a process depends upon how the process is carried out, work is *not* a state function. Thus, we write
$$\int_{1}^{2} \delta w = w \qquad \text{(not } \Delta w \text{ or } w_2 - w_1\text{)}$$
(19.5)
It makes no sense at all to write $w_2$ , $w_1$ , $w_2-w_1$ , or $\Delta w$ . The val... | {
"Header 1": "**13–3.** Vibration–Rotation Interaction Accounts for the Unequal Spacing of the Lines in the *P* and *R* Branches of a Vibration-Rotation Spectrum",
"Header 2": "**19–3.** The First Law of Thermodynamics Says the Energy Is a State Function",
"token_count": 615,
"source_pdf": "datasets/websources... |
Not only are work and heat not state functions, but we can prove that even reversible work and reversible heat are not state functions by a direct calculation. Consider the three paths, depicted in Figure 19.5, that occur between the same initial and final states, $P_1$ , $V_1$ , $T_1$ and $P_2$ , $V_2$ , $T_1$ ... | {
"Header 1": "**13–3.** Vibration–Rotation Interaction Accounts for the Unequal Spacing of the Lines in the *P* and *R* Branches of a Vibration-Rotation Spectrum",
"Header 2": "**19–4.** An Adiabatic Process Is a Process in Which No Energy as Heat Is Transferred",
"token_count": 1861,
"source_pdf": "datasets/w... |
$$\Delta U_{\rm D+E} = \Delta U_{\rm D} + \Delta U_{\rm E} = \int_{T_{\rm I}}^{T_{\rm 3}} C_{V}(T)dT + \int_{T_{\rm 3}}^{T_{\rm 1}} C_{V}(T)dT = 0$$
$$w_{\text{rev},D+E} = w_{\text{rev},D} + w_{\text{rev},E} = -P_1(V_2 - V_1)$$
and
$$q_{\text{rev},D+E} = q_{\text{rev},D} + q_{\text{rev},E} = P_1(V_2 - V_1)$$
... | {
"Header 1": "**13–3.** Vibration–Rotation Interaction Accounts for the Unequal Spacing of the Lines in the *P* and *R* Branches of a Vibration-Rotation Spectrum",
"Header 2": "**19–4.** An Adiabatic Process Is a Process in Which No Energy as Heat Is Transferred",
"token_count": 228,
"source_pdf": "datasets/we... |
Path B in Figure 19.5 represents the reversible adiabatic expansion of an ideal gas from $T_1$ , $V_1$ to $T_2$ , $V_2$ . As the figure suggests, $T_2 < T_1$ , which means that the gas cools during a (reversible) adiabatic expansion. We can determine the final temperature $T_2$ for this process. For an adiabati... | {
"Header 1": "**13–3.** Vibration–Rotation Interaction Accounts for the Unequal Spacing of the Lines in the *P* and *R* Branches of a Vibration-Rotation Spectrum",
"Header 2": "**19–5.** The Temperature of a Gas Decreases in a Reversible Adiabatic Expansion",
"token_count": 1456,
"source_pdf": "datasets/websou... |
Let's go back to Equation 17.18 for the average energy of a macroscopic system,
$$U = \sum_{j} p_{j}(N, V, \beta) E_{j}(N, V)$$
(19.24)
with
$$p_{j}(N, V, \beta) = \frac{e^{-\beta E_{j}(N, V)}}{Q(N, V, \beta)}$$
(19.25)
Equation 19.24 represents the average energy of an equilibrium system that has the variables... | {
"Header 1": "**13–3.** Vibration–Rotation Interaction Accounts for the Unequal Spacing of the Lines in the *P* and *R* Branches of a Vibration-Rotation Spectrum",
"Header 2": "**19–5.** The Temperature of a Gas Decreases in a Reversible Adiabatic Expansion",
"Header 3": "**19-6.** Work and Heat Have a Simple Mo... |
For a reversible process in which the only work involved is pressure-volume work, the first law tells us that
$$\Delta U = q + w = q - \int_{V_1}^{V_2} P dV$$
(19.32)
If the process is carried out at constant volume, then $V_1 = V_2$ and
$$\Delta U = q_{\nu} \tag{19.33}$$
where the subscript V on q emphasizes... | {
"Header 1": "**13–3.** Vibration–Rotation Interaction Accounts for the Unequal Spacing of the Lines in the *P* and *R* Branches of a Vibration-Rotation Spectrum",
"Header 2": "**19–7.** The Enthalpy Change Is Equal to the Energy Transferred as Heat in a Constant-Pressure Process Involving Only P-V Work",
"token... |
Recall that heat capacity is defined as the energy as heat required to raise the temperature of a substance by one degree. The heat capacity also depends upon the temperature T. Because the energy required to raise the temperature of a substance by one kelvin depends upon the amount of substance, heat capacity is an *e... | {
"Header 1": "**13–3.** Vibration–Rotation Interaction Accounts for the Unequal Spacing of the Lines in the *P* and *R* Branches of a Vibration-Rotation Spectrum",
"Header 2": "**19–7.** The Enthalpy Change Is Equal to the Energy Transferred as Heat in a Constant-Pressure Process Involving Only P-V Work",
"Heade... |
By integrating Equation 19.40, we can calculate the difference in the enthalpy of a substance that does not change phase between two temperatures:
$$H(T_2) - H(T_1) = \int_{T_1}^{T_2} C_P(T) dT$$
(19.44)
If we letT= 0 K, we have
$$H(T) - H(0) = \int_0^T C_P(T')dT'$$
(19.45)
[Notice that we have written the inte... | {
"Header 1": "**13–3.** Vibration–Rotation Interaction Accounts for the Unequal Spacing of the Lines in the *P* and *R* Branches of a Vibration-Rotation Spectrum",
"Header 2": "**19-9.** Relative Enthalpies Can Be Determined from Heat Capacity Data and Heats of Transition",
"token_count": 883,
"source_pdf": "d... |
Because most chemical reactions take place at constant pressure (open to the atmosphere), the enthalpy change associated with chemical reactions, ~,H, (the subscript r indicates that the enthalpy change is for a chemical reaction) plays a central role in *thermochemistry,* which is the branch of thermodynamics that con... | {
"Header 1": "**13–3.** Vibration–Rotation Interaction Accounts for the Unequal Spacing of the Lines in the *P* and *R* Branches of a Vibration-Rotation Spectrum",
"Header 2": "**19-10.** Enthalpy Changes for Chemical Equations Are Additive",
"token_count": 1678,
"source_pdf": "datasets/websources/biochem/F814... |
The molar enthalpies of combustion of isobutane and n-butane are -2871 kJ ·mol~ <sup>1</sup>and -2878 kJ-mol~ , respectively at 298K and one atm. Calculate /1rH for the conversion of one mole of n-butane to one mole of isobutane.
S 0 L UTI 0 N : The equations for the two combustion reactions are
$$n-C_4H_{10}(g) + ... | {
"Header 1": "**13–3.** Vibration–Rotation Interaction Accounts for the Unequal Spacing of the Lines in the *P* and *R* Branches of a Vibration-Rotation Spectrum",
"Header 2": "**19-10.** Enthalpy Changes for Chemical Equations Are Additive",
"Header 3": "**EXAMPLE 19-9**",
"token_count": 475,
"source_pdf": ... |
The enthalpy change of a chemical reaction, *!'3.rH,* depends upon the number of moles of the reactants. Recently, the physical chemistry division of the International Union of Pure and Applied Chemistry (IUPAC) has proposed a systematic procedure for tabulating reaction enthalpies. The *standard reaction enthalpy* of ... | {
"Header 1": "**13–3.** Vibration–Rotation Interaction Accounts for the Unequal Spacing of the Lines in the *P* and *R* Branches of a Vibration-Rotation Spectrum",
"Header 2": "**19-10.** Enthalpy Changes for Chemical Equations Are Additive",
"Header 3": "**19-11.** Heats of Reactions Can Be Calculated from Tabu... |
S 0 L UTI 0 N : The chemical equations for the three combustion reactions are as follows:
$$C(s) + O_2(g) \longrightarrow CO_2(g) \qquad \Delta_c H^{\circ}(1) = -393.51 \text{ kJ} \cdot \text{mol}^{-1}$$
(1)
$$H_2(g) + \frac{1}{2} O_2(g) \longrightarrow H_2O(l) \qquad \Delta_c H^{\circ}(2) = -285.83 \text{ kJ} \c... | {
"Header 1": "**13–3.** Vibration–Rotation Interaction Accounts for the Unequal Spacing of the Lines in the *P* and *R* Branches of a Vibration-Rotation Spectrum",
"Header 2": "**19-10.** Enthalpy Changes for Chemical Equations Are Additive",
"Header 3": "**19-11.** Heats of Reactions Can Be Calculated from Tabu... |
| b<br>S<br>t<br>u<br>s<br>a<br>n<br>c<br>e | l<br>F<br>o<br>r<br>m<br>u<br>a | 1<br>!:<br>!<br>H<br>/<br>k<br>J<br>l<br>r<br>o<br>m<br>o<br>·<br>- |
|-------------------------------------------------------------------------------------------... | {
"Header 1": "**13–3.** Vibration–Rotation Interaction Accounts for the Unequal Spacing of the Lines in the *P* and *R* Branches of a Vibration-Rotation Spectrum",
"Header 2": "**19-10.** Enthalpy Changes for Chemical Equations Are Additive",
"Header 3": "**19-11.** Heats of Reactions Can Be Calculated from Tabu... |
We can calculate L}.,H in two steps, as shown in the following diagram:
First, we decompose compounds A and B into their constituent elements (step 1); and then we combine the elements to form the compounds Y and Z (step 2). In the first step, we have
$$\Delta_{\mathbf{r}}H(1) = -a\De... | {
"Header 1": "**13–3.** Vibration–Rotation Interaction Accounts for the Unequal Spacing of the Lines in the *P* and *R* Branches of a Vibration-Rotation Spectrum",
"Header 2": "**19-10.** Enthalpy Changes for Chemical Equations Are Additive",
"Header 3": "**19-11.** Heats of Reactions Can Be Calculated from Tabu... |
Up to now, we have calculated reaction entha1pies at 25oC. We will see in this section that we can calculate ~,H at other temperatures if we have sufficient heat-capacity data. Consider the general reaction
$$aA + bB \longrightarrow yY + zZ$$
We can express ~,H at a temperature T*2*in the form
$$\Delta_{r}H(T_{2}... | {
"Header 1": "**13–3.** Vibration–Rotation Interaction Accounts for the Unequal Spacing of the Lines in the *P* and *R* Branches of a Vibration-Rotation Spectrum",
"Header 2": "**19-10.** Enthalpy Changes for Chemical Equations Are Additive",
"Header 3": "**19-12.** The Temperature Dependence of *!),.rH* Is Give... |
$$\begin{split} &C_P^{\circ}(\mathrm{H_2})/\mathrm{J}\cdot\mathrm{K^{-1}}\cdot\mathrm{mol^{-1}} = 29.07 - (0.837\times10^{-3}~\mathrm{K^{-1}})T + (2.012\times10^{-6}~\mathrm{K^{-2}})T^2 \\ &C_P^{\circ}(\mathrm{N_2})/\mathrm{J}\cdot\mathrm{K^{-1}}\cdot\mathrm{mol^{-1}} = 26.98 + (5.912\times10^{-3}~\mathrm{K^{-1}})T -... | {
"Header 1": "**13–3.** Vibration–Rotation Interaction Accounts for the Unequal Spacing of the Lines in the *P* and *R* Branches of a Vibration-Rotation Spectrum",
"Header 2": "**19-10.** Enthalpy Changes for Chemical Equations Are Additive",
"Header 3": "**19-12.** The Temperature Dependence of *!),.rH* Is Give... |
- **19-1.** Suppose that a 10-kg mass of iron at 20°C is dropped from a height of 100 meters. What is the kinetic energy of the mass just before it hits the ground? What is its speed? What would be the final temperature of the mass if all its kinetic energy at impact is transformed into internal energy? Take the molar ... | {
"Header 1": "**13–3.** Vibration–Rotation Interaction Accounts for the Unequal Spacing of the Lines in the *P* and *R* Branches of a Vibration-Rotation Spectrum",
"Header 2": "**Problems**",
"token_count": 2021,
"source_pdf": "datasets/websources/biochem/F814BC5915875384820.pdf"
} |
Show that
$$\left(\frac{T_2}{T_1}\right)^{3/2} = \frac{\overline{V}_1 - b}{\overline{V}_2 - b}$$
for a reversible, adiabatic expansion of a monatomic gas that obeys the equation of state $P(\overline{V} - b) = RT$ . Extend this result to the case of a diatomic gas.
19-17. Show that
$$\frac{T_2}{T_1} = \left(\f... | {
"Header 1": "**13–3.** Vibration–Rotation Interaction Accounts for the Unequal Spacing of the Lines in the *P* and *R* Branches of a Vibration-Rotation Spectrum",
"Header 2": "**Problems**",
"token_count": 1766,
"source_pdf": "datasets/websources/biochem/F814BC5915875384820.pdf"
} |
- **19-30.** Given that $(\partial U/\partial V)_T = 0$ for an ideal gas, prove that $(\partial H/\partial V)_T = 0$ for an ideal gas.
- **19-31.** Given that $(\partial U/\partial V)_T = 0$ for an ideal gas, prove that $(\partial C_V/\partial V)_T = 0$ for an ideal gas.
- **19-32.** Show that $C_P C_V = nR$... | {
"Header 1": "**13–3.** Vibration–Rotation Interaction Accounts for the Unequal Spacing of the Lines in the *P* and *R* Branches of a Vibration-Rotation Spectrum",
"Header 2": "**Problems**",
"token_count": 1623,
"source_pdf": "datasets/websources/biochem/F814BC5915875384820.pdf"
} |
$$\begin{split} C_P^{\circ}[\mathrm{CO}(\mathrm{g})]/R &= 3.231 + (8.379 \times 10^{-4} \; \mathrm{K}^{-1})T - (9.86 \times 10^{-8} \; \mathrm{K}^{-2})T^2 \\ C_P^{\circ}[\mathrm{H}_2(\mathrm{g})]/R &= 3.496 + (1.006 \times 10^{-4} \; \mathrm{K}^{-1})T + (2.42 \times 10^{-7} \; \mathrm{K}^{-2})T^2 \\ C_P^{\circ}[\math... | {
"Header 1": "**13–3.** Vibration–Rotation Interaction Accounts for the Unequal Spacing of the Lines in the *P* and *R* Branches of a Vibration-Rotation Spectrum",
"Header 2": "**Problems**",
"token_count": 2026,
"source_pdf": "datasets/websources/biochem/F814BC5915875384820.pdf"
} |
Use the First Law of Thermodynamics to show that
$$U_2 + P_2 V_2 = U_1 + P_1 V_1$$
or that $\Delta H = 0$ for a Joule-Thomson expansion.
Starting with
$$dH = \left(\frac{\partial H}{\partial P}\right)_T dP + \left(\frac{\partial H}{\partial T}\right)_P dT$$
show that
$$\left(\frac{\partial T}{\partial P}\... | {
"Header 1": "**13–3.** Vibration–Rotation Interaction Accounts for the Unequal Spacing of the Lines in the *P* and *R* Branches of a Vibration-Rotation Spectrum",
"Header 2": "**Problems**",
"token_count": 741,
"source_pdf": "datasets/websources/biochem/F814BC5915875384820.pdf"
} |
In the next chapter, we will learn about entropy, a thermodynamic state function that has a molecular interpretation of being a measure of the disorder of a system. In doing so, we will have to put the idea of the disorder of a system on a quantitative basis. A problem we will encounter is that of determining how many ... | {
"Header 1": "**13–3.** Vibration–Rotation Interaction Accounts for the Unequal Spacing of the Lines in the *P* and *R* Branches of a Vibration-Rotation Spectrum",
"Header 2": "**THE BINOMIAL DISTRIBUTION AND STIRLING'S APPROXIMATION**",
"token_count": 2017,
"source_pdf": "datasets/websources/biochem/F814BC591... |
#### **EXAMPLE J-3**
A more refined version of Stirling's approximation (one we will *not* have to use in the next chapter) says that
$$\ln N! = N \ln N - N + \ln(2\pi N)^{1/2}$$
Use this version of Stirling's approximation to calculate $\ln N!$ for N=10 and compare the relative error with that in Table J.1. ... | {
"Header 1": "**13–3.** Vibration–Rotation Interaction Accounts for the Unequal Spacing of the Lines in the *P* and *R* Branches of a Vibration-Rotation Spectrum",
"Header 2": "**THE BINOMIAL DISTRIBUTION AND STIRLING'S APPROXIMATION**",
"token_count": 927,
"source_pdf": "datasets/websources/biochem/F814BC5915... |
- **J-1.** Use Equation J.3 to write the expansion of $(1+x)^5$ . Use Equation J.4 to do the same thing.
- **J-2.** Use Equation J.6 to write out the expression for $(x + y + z)^2$ . Compare your result to the one that you obtain by multiplying (x + y + z) by (x + y + z).
- **J-3.** Use Equation J.6 to write out the ... | {
"Header 1": "**13–3.** Vibration–Rotation Interaction Accounts for the Unequal Spacing of the Lines in the *P* and *R* Branches of a Vibration-Rotation Spectrum",
"Header 2": "**Problems**",
"token_count": 1420,
"source_pdf": "datasets/websources/biochem/F814BC5915875384820.pdf"
} |
For years, scientists wondered why some reactions or processes proceed spontaneously and others do not. We all know that under the right conditions iron rusts, and that objects do not spontaneously unrust. We all know that hydrogen and oxygen react explosively to form water but that an input of energy by means of elect... | {
"Header 1": "**13–3.** Vibration–Rotation Interaction Accounts for the Unequal Spacing of the Lines in the *P* and *R* Branches of a Vibration-Rotation Spectrum",
"Header 2": "**20-1.** The Change of Energy Alone Is Not Sufficient to Determine the Direction of a Spontaneous Process",
"token_count": 922,
"sour... |
If we examine the above processes from a microscopic or molecular point of view, we see that each one involves an increase in disorder or randomness of the system. For example, in Figure 20.1, the gas molecules in the final state are able to move over a volume that is twice as large as in the initial state. In a sense,... | {
"Header 1": "**13–3.** Vibration–Rotation Interaction Accounts for the Unequal Spacing of the Lines in the *P* and *R* Branches of a Vibration-Rotation Spectrum",
"Header 2": "**20-1.** The Change of Energy Alone Is Not Sufficient to Determine the Direction of a Spontaneous Process",
"Header 3": "**20-2.** None... |
In the previous chapter, we calculated the reversible work and reversible heat for two processes that take place between the same initial and final states (Figure 20.3). The first process involved a reversible isothermal expansion of an ideal gas from *P*1, V1, T1 to P*<sup>2</sup> ,* V*<sup>2</sup> ,* T1 (path A). For... | {
"Header 1": "**13–3.** Vibration–Rotation Interaction Accounts for the Unequal Spacing of the Lines in the *P* and *R* Branches of a Vibration-Rotation Spectrum",
"Header 2": "**20-1.** The Change of Energy Alone Is Not Sufficient to Determine the Direction of a Spontaneous Process",
"Header 3": "**20-3.** Unli... |
But this is given by $P_1V = nRT$ , so
$$\begin{split} \Delta S_D &= \int_{T_1}^{T_3} \frac{C_V(T)}{T} dT + nR \int_{V_1}^{V_2} \frac{dV}{V} \\ &= \int_{T_1}^{T_3} \frac{C_V(T)}{T} dT + nR \ln \frac{V_2}{V_1} \end{split}$$
For path E, $\delta w_{\text{rev}} = 0$ , and using Equation 20.12 for $\delta q_{\text{re... | {
"Header 1": "**13–3.** Vibration–Rotation Interaction Accounts for the Unequal Spacing of the Lines in the *P* and *R* Branches of a Vibration-Rotation Spectrum",
"Header 2": "**20-1.** The Change of Energy Alone Is Not Sufficient to Determine the Direction of a Spontaneous Process",
"Header 3": "**20-3.** Unli... |
We all know that energy as heat will flow spontaneously from a region of high temperature to a region of low temperature. Let's investigate the role entropy plays in this
process. Consider the two-compartment system shown in Figure 20.4, where parts A and B are large one-component systems. Both systems are at equilib... | {
"Header 1": "**20–4.** The Second Law of Thermodynamics States That the Entropy of an Isolated System Increases as a Result of a Spontaneous Process",
"token_count": 2043,
"source_pdf": "datasets/websources/biochem/F814BC5915875384820.pdf"
} |
Then the system is allowed to interact with its surroundings and is brought back to state 1 by some reversible path. Because entropy is a state function, $\Delta S = 0$ for a cyclic process.
Consider a cyclic process (Figure 20.6) in which a system is first isolated and undergoes an irreversible process from state ... | {
"Header 1": "**20–4.** The Second Law of Thermodynamics States That the Entropy of an Isolated System Increases as a Result of a Spontaneous Process",
"token_count": 399,
"source_pdf": "datasets/websources/biochem/F814BC5915875384820.pdf"
} |
In this section, we will discuss the molecular interpretation of entropy more quantitatively than we have up to now. We have shown that entropy is a state function that is related to the disorder of a system. Disorder can be expressed in a number of ways, but the way that has turned out to be the most useful is the fol... | {
"Header 1": "**20–4.** The Second Law of Thermodynamics States That the Entropy of an Isolated System Increases as a Result of a Spontaneous Process",
"Header 2": "**20–5.** The Most Famous Equation of Statistical Thermodynamics Is $S = k_{\\rm B} \\ln W$",
"token_count": 2005,
"source_pdf": "datasets/websour... |
Then $\Delta_{mix}S$ is given by
$$\begin{split} \Delta_{\text{mix}} S &= S_{\text{mixture}} - S_1 - S_2 \\ &= k_{\text{B}} \ln \frac{\Omega_{\text{mixture}}}{\Omega_1 \Omega_2} \end{split}$$
where 1 and 2 refer to $N_2(g)$ and $Br_2(g)$ , respectively. The quantities $\Omega_1$ and $\Omega_2$ are given by... | {
"Header 1": "**20–4.** The Second Law of Thermodynamics States That the Entropy of an Isolated System Increases as a Result of a Spontaneous Process",
"Header 2": "**20–5.** The Most Famous Equation of Statistical Thermodynamics Is $S = k_{\\rm B} \\ln W$",
"token_count": 773,
"source_pdf": "datasets/websourc... |
The discussion so far has been fairly abstract, and it will be helpful at this point to illustrate the change of entropy in a spontaneous process by means of some calculations involving an ideal gas for simplicity. First, let's consider the situation in Figure 20.1, in which an ideal gas at T and $V_1$ is allowed to ... | {
"Header 1": "**20–4.** The Second Law of Thermodynamics States That the Entropy of an Isolated System Increases as a Result of a Spontaneous Process",
"Header 2": "**20–6.** We Must Always Devise a Reversible Process to Calculate Entropy Changes",
"token_count": 1788,
"source_pdf": "datasets/websources/bioche... |
For nitrogen, we have (using Equation 20.28)
$$\Delta S_{\rm N_2} = n_{\rm N_2} R \ln \frac{V_{\rm N_2} + V_{\rm Br_2}}{V_{\rm N_2}} = -n_{\rm N_2} R \ln \frac{V_{\rm N_2}}{V_{\rm N_2} + V_{\rm Br_2}}$$
and for bromine,
$$\Delta S_{\mathrm{Br_2}} = n_{\mathrm{Br_2}} R \ln \frac{V_{\mathrm{N_2}} + V_{\mathrm{Br_2}... | {
"Header 1": "**20–4.** The Second Law of Thermodynamics States That the Entropy of an Isolated System Increases as a Result of a Spontaneous Process",
"Header 2": "**20–6.** We Must Always Devise a Reversible Process to Calculate Entropy Changes",
"token_count": 1679,
"source_pdf": "datasets/websources/bioche... |
SOLUTION: As usual, we start with Equation 20.19
$$dS = \frac{\delta q_{\text{rev}}}{T}$$
In this case, $\delta q_{\text{rev}} = \overline{C}_P(T)dT$ , so
$$\Delta \overline{S} = \int_{T_1}^{T_2} \frac{\overline{C}_P(T)}{T} dT$$
Using the given expression for $\overline{C}_{p}(T)$ , we have
$$\Delta \over... | {
"Header 1": "**20–4.** The Second Law of Thermodynamics States That the Entropy of an Isolated System Increases as a Result of a Spontaneous Process",
"Header 2": "**20–6.** We Must Always Devise a Reversible Process to Calculate Entropy Changes",
"token_count": 451,
"source_pdf": "datasets/websources/biochem... |
The concept of entropy and the Second Law of Thermodynamics was first developed by a French engineer named Sadi Carnot in the 1820s in a study of the efficiency of the newly developed steam engines and other types of heat engines. Although primarily of historical interest to chemists, the result of Carnot's analysis is... | {
"Header 1": "**20–4.** The Second Law of Thermodynamics States That the Entropy of an Isolated System Increases as a Result of a Spontaneous Process",
"Header 2": "**20–7.** Thermodynamics Gives Us Insight into the Conversion of Heat into Work",
"token_count": 901,
"source_pdf": "datasets/websources/biochem/F... |
We presented the equation $S = k_B \ln W$ in Section 20–5. This equation can be used as the starting point to derive most of the important results of statistical thermodynamics. For example, we can use it to derive an expression for the entropy in terms of the system partition function, $Q(N, V, \beta)$ , as we have... | {
"Header 1": "**20–4.** The Second Law of Thermodynamics States That the Entropy of an Isolated System Increases as a Result of a Spontaneous Process",
"Header 2": "**20–7.** Thermodynamics Gives Us Insight into the Conversion of Heat into Work",
"Header 3": "20-8. Entropy Can Be Expressed in Terms of a Partitio... |
S 0 L UTI 0 N : At 298.2 K and one bar,
$$\begin{split} \frac{N_{\rm A}}{\overline{V}} &= \frac{N_{\rm A}P}{RT} \\ &= \frac{(6.022 \times 10^{23} \ {\rm mol}^{-1})(1 \ {\rm bar})}{(0.08314 \ {\rm L} \cdot {\rm bar} \cdot {\rm K}^{-1} \cdot {\rm mol}^{-1})(298.2 \ {\rm K})} \\ &= 2.429 \times 10^{22} \ {\rm L}^{-1} ... | {
"Header 1": "**20–4.** The Second Law of Thermodynamics States That the Entropy of an Isolated System Increases as a Result of a Spontaneous Process",
"Header 2": "**20–7.** Thermodynamics Gives Us Insight into the Conversion of Heat into Work",
"Header 3": "20-8. Entropy Can Be Expressed in Terms of a Partitio... |
In this last section, we will show that Equation 20.24, or its equivalent, Equation 20.40, is consistent with our thermodynamic definition of the entropy. As a bonus, we will finally prove that *fJ* = 1 I *k8 T.*
If we differentiate Equation 20.40 with respect to p., we get <sup>J</sup>
$$dS = -k_{\rm B} \sum_{i} \... | {
"Header 1": "**20–4.** The Second Law of Thermodynamics States That the Entropy of an Isolated System Increases as a Result of a Spontaneous Process",
"Header 2": "**20-9.** The Molecular FormulaS= *kB* **ln** *W* Is Analogous to the Thermodynamic Formula *dS* = *8qrevl T*",
"token_count": 542,
"source_pdf": ... |
20-1. Show that
$$\oint dY = 0$$
if *Y* is a state function.
- 20-2. Let *z* = *z(x,* y) and *dz* = *xydx* + *idy.* Although *dz* is not an exact differential (why not?), what combination of *dz* and *x* and/or *y* is an exact differential?
- 20-3. Use the criterion developed in MathChapter H to prove that 8qrev ... | {
"Header 1": "**20–4.** The Second Law of Thermodynamics States That the Entropy of an Isolated System Increases as a Result of a Spontaneous Process",
"Header 2": "Problems",
"token_count": 1984,
"source_pdf": "datasets/websources/biochem/F814BC5915875384820.pdf"
} |
- **20-18.** Vaporization at the normal boiling point $(T_{\text{vap}})$ of a substance (the boiling point at one atm) can be regarded as a reversible process because if the temperature is decreased infinitesimally below $T_{\text{vap}}$ , all the vapor will condense to liquid, whereas if it is increased infinites... | {
"Header 1": "**20–4.** The Second Law of Thermodynamics States That the Entropy of an Isolated System Increases as a Result of a Spontaneous Process",
"Header 2": "Problems",
"token_count": 2029,
"source_pdf": "datasets/websources/biochem/F814BC5915875384820.pdf"
} |
**20-33.** Calculate the change in entropy if one mole of SO<sub>2</sub>(g) at 300 K and 1.00 bar is heated to 1000 K and its pressure is decreased to 0.010 bar. Take the molar heat capacity of SO<sub>2</sub>(g) to be
$$\overline{C}_P(T)/R = 7.871 - \frac{1454.6 \text{ K}}{T} + \frac{160351 \text{ K}^2}{T^2}$$
**... | {
"Header 1": "**20–4.** The Second Law of Thermodynamics States That the Entropy of an Isolated System Increases as a Result of a Spontaneous Process",
"Header 2": "Problems",
"token_count": 1989,
"source_pdf": "datasets/websources/biochem/F814BC5915875384820.pdf"
} |
We start with the First Law of Thermodynamics for a reversible process:
$$dU = \delta q_{\rm rev} + \delta w_{\rm rev}$$
Using the fact that *8qrev* = *TdS* and *8wrev* = *-PdV,* we obtain a combination of the First and Second Laws of Thermodynamics:
$$dU = TdS - PdV (21.1)$$
We can derive a number of relations... | {
"Header 1": "**20–4.** The Second Law of Thermodynamics States That the Entropy of an Isolated System Increases as a Result of a Spontaneous Process",
"Header 2": "**21-1.** Entropy Increases with Increasing Temperature",
"token_count": 1283,
"source_pdf": "datasets/websources/biochem/F814BC5915875384820.pdf"... |
Let's discuss S(0 K) first. Around the turn of the century, the German chemist Walther Nernst, after studying numerous chemical reactions, postulated that $\Delta_r S \to 0$ as $T \to 0$ . Nernst did not make any statement concerning the entropy of any particular substance at 0 K, only that all pure crystalline subs... | {
"Header 1": "**20–4.** The Second Law of Thermodynamics States That the Entropy of an Isolated System Increases as a Result of a Spontaneous Process",
"Header 2": "**21–2.** The Third Law of Thermodynamics Says That the Entropy of a Perfect Crystal Is Zero at 0 K",
"token_count": 848,
"source_pdf": "datasets/... |
We made a tacit assumption when we wrote Equation 21.14; we assumed that there is no phase transition between 0 and T. Suppose there is such a transition at $T_{\rm trs}$ between 0 and T. We can calculate the entropy change upon the phase transition, $\Delta_{\rm trs} S$ , by using the equation
$$\Delta_{\rm trs}S... | {
"Header 1": "**20–4.** The Second Law of Thermodynamics States That the Entropy of an Isolated System Increases as a Result of a Spontaneous Process",
"Header 2": "**21–2.** The Third Law of Thermodynamics Says That the Entropy of a Perfect Crystal Is Zero at 0 K",
"Header 3": "**21–3.** $\\Delta_{\\text{trs}}S... |
It has been shown experimentally and theoretically that $C_P^s(T) \to T^3$ as $T \to 0$ for most nonmetallic crystals ( $C_P^s$ for metallic crystals goes as $aT + bT^3$ as $T \to 0$ , where a and b are constants). This $T^3$ temperature dependence is valid from 0 K to about 15 K and is called the *Debye* $T... | {
"Header 1": "**20–4.** The Second Law of Thermodynamics States That the Entropy of an Isolated System Increases as a Result of a Spontaneous Process",
"Header 2": "**21–4.** The Third Law of Thermodynamics Asserts That $C_P \\to 0$ as $T \\to 0$",
"token_count": 801,
"source_pdf": "datasets/websources/biochem... |
Given suitable heat capacity data and enthalpies of transition and transition temperatures, we can use Equation 21.17 to calculate entropies based on the convention of setting S(O K) = 0. Such entropies are called third-law entropies, or practical absolute entropies. Table 21.1 gives the entropy of N2 (g) at 298.15 K. ... | {
"Header 1": "**20–4.** The Second Law of Thermodynamics States That the Entropy of an Isolated System Increases as a Result of a Spontaneous Process",
"Header 2": "**21–4.** The Third Law of Thermodynamics Asserts That $C_P \\to 0$ as $T \\to 0$",
"Header 3": "**21-5.** Practical Absolute Entropies Can Be Deter... |
Recall from Section 20-8 that the entropy can be written as (Equation 20.43)
$$S = k_{\rm B} \ln Q + k_{\rm B} T \left( \frac{\partial \ln Q}{\partial T} \right)_{N,V} \tag{21.19}$$
where *Q(N, V,* T) is the system partition function
$$Q(N, V, T) = \sum_{j} e^{-E_{j}(N, V)/k_{\rm B}T}$$
(21.20)
Equation 21.19 i... | {
"Header 1": "**20–4.** The Second Law of Thermodynamics States That the Entropy of an Isolated System Increases as a Result of a Spontaneous Process",
"Header 2": "**21–4.** The Third Law of Thermodynamics Asserts That $C_P \\to 0$ as $T \\to 0$",
"Header 3": "**21-6.** Practical Absolute Entropies of Gases Can... |
The necessary parameters are $\Theta_{\rm rot} = 2.88~{\rm K},~\Theta_{\rm vib} = 3374~{\rm K},~{\rm and}~g_{e1} = 1.$ At 298.15 K and one bar, the various factors are
$$\left(\frac{2\pi M k_{\rm B} T}{h^2}\right)^{3/2} = \left[\frac{2\pi (4.653 \times 10^{-26} \text{ kg})(1.3807 \times 10^{-23} \text{ J} \cdot \te... | {
"Header 1": "**20–4.** The Second Law of Thermodynamics States That the Entropy of an Isolated System Increases as a Result of a Spontaneous Process",
"Header 2": "**21–4.** The Third Law of Thermodynamics Asserts That $C_P \\to 0$ as $T \\to 0$",
"Header 3": "**21-6.** Practical Absolute Entropies of Gases Can... |
We substitute Equation 21.25 into Equation 21.27 to obtain
$$\begin{split} \frac{S^{\circ}}{R} &= 1 + \ln \left[ \left( \frac{2\pi M k_{\rm B} T}{h^2} \right)^{3/2} \frac{\overline{V}}{N_{\rm A}} \right] + \ln \left( \frac{T}{\sigma \Theta_{\rm rot}} \right) \\ &- \sum_{j=1}^{4} \frac{\Theta_{\rm vib, j}}{2T} - \sum_... | {
"Header 1": "**20–4.** The Second Law of Thermodynamics States That the Entropy of an Isolated System Increases as a Result of a Spontaneous Process",
"Header 2": "**21–4.** The Third Law of Thermodynamics Asserts That $C_P \\to 0$ as $T \\to 0$",
"Header 3": "**21-6.** Practical Absolute Entropies of Gases Can... |
Let's look at the standard molar entropy values in Table 21.2 and try to determine some trends. First, notice that the standard molar entropies of the gaseous substances are the largest, and the standard molar entropies of the solid substances are the smallest. These values reflect the fact that solids are more ordered... | {
"Header 1": "**20–4.** The Second Law of Thermodynamics States That the Entropy of an Isolated System Increases as a Result of a Spontaneous Process",
"Header 2": "**21–7.** The Values of Standard Molar Entropies Depend Upon Molecular Mass and Molecular Structure",
"token_count": 1701,
"source_pdf": "datasets... |
The value of $\Delta \overline{S}$ for this first step is (Equation 21.7)
$$\begin{split} \Delta \overline{S}_1 &= \overline{S}^1 (332.0 \text{ K}) - \overline{S}^1 (298.15 \text{ K}) = \overline{C}_P^1 \ln \frac{T_2}{T_1} \\ &= (75.69 \text{ J} \cdot \text{K}^{-1} \cdot \text{mol}^{-1}) \ln \frac{332.0 \text{ K}}{... | {
"Header 1": "**20–4.** The Second Law of Thermodynamics States That the Entropy of an Isolated System Increases as a Result of a Spontaneous Process",
"Header 2": "**21–7.** The Values of Standard Molar Entropies Depend Upon Molecular Mass and Molecular Structure",
"token_count": 1016,
"source_pdf": "datasets... |
Table 21.4 compares calculated values of the molar entropies of several polyatomic gases with those measured calorimetrically. Note again, that the agreement with experiment is quite good. In fact, calculated values of the entropy are often more accurate than measured values, provided sophisticated enough spectroscopic... | {
"Header 1": "**20–4.** The Second Law of Thermodynamics States That the Entropy of an Isolated System Increases as a Result of a Spontaneous Process",
"Header 2": "**21–8.** The Spectroscopic Entropies of a Few Substances Do Not Agree with the Calorimetric Entropies",
"token_count": 1059,
"source_pdf": "datas... |
One of the most important uses of tables of standard molar entropies is for the calculation of entropy changes of chemical reactions. These changes are calculated in much the same way we calculated standard enthalpy changes of reactions from standard molar enthalpies of formation in Chapter 19. For the general reaction... | {
"Header 1": "**20–4.** The Second Law of Thermodynamics States That the Entropy of an Isolated System Increases as a Result of a Spontaneous Process",
"Header 2": "**21–8.** The Spectroscopic Entropies of a Few Substances Do Not Agree with the Calorimetric Entropies",
"Header 3": "**21-9.** Standard Entropies C... |
- **21-1.** Form the total derivative of H as a function of T and P and equate the result to dH in Equation 21.6 to derive Equations 21.7 and 21.8.
- **21-2.** The molar heat capacity of $H_2O(1)$ has an approximately constant value of $\overline{C}_p = 75.4 \text{ J} \cdot \text{K}^{-1} \cdot \text{mol}^{-1}$ from... | {
"Header 1": "**20–4.** The Second Law of Thermodynamics States That the Entropy of an Isolated System Increases as a Result of a Spontaneous Process",
"Header 2": "**21–8.** The Spectroscopic Entropies of a Few Substances Do Not Agree with the Calorimetric Entropies",
"Header 3": "**Problems**",
"token_count"... |
$$\begin{split} C_P^\circ[\mathrm{N}_2(\mathrm{s}_1)]/R &= -0.03165 + (0.05460~\mathrm{K}^{-1})T + (3.520\times 10^{-3}~\mathrm{K}^{-2})T^2 \\ &- (2.064\times 10^{-5}~\mathrm{K}^{-3})T^3 \\ &- 10~\mathrm{K} \leq T \leq 35.61~\mathrm{K} \\ \\ C_P^\circ[\mathrm{N}_2(\mathrm{s}_2)]/R &= -0.1696 + (0.2379~\mathrm{K}^{-1}... | {
"Header 1": "**20–4.** The Second Law of Thermodynamics States That the Entropy of an Isolated System Increases as a Result of a Spontaneous Process",
"Header 2": "**21–8.** The Spectroscopic Entropies of a Few Substances Do Not Agree with the Calorimetric Entropies",
"Header 3": "**Problems**",
"token_count"... |
$$\begin{split} C_p^\circ[\mathrm{C_3H_6(s)}]/R &= -1.921 + (0.1508~\mathrm{K^{-1}})T - (9.670\times10^{-4}~\mathrm{K^{-2}})T^2 \\ &\quad + (2.694\times10^{-6}~\mathrm{K^{-3}})T^3 \\ &\quad 15~\mathrm{K} \leq T \leq 145.5~\mathrm{K} \\ \\ C_p^\circ[\mathrm{C_3H_6(l)}]/R &= 5.624 + (4.493\times10^{-2}~\mathrm{K^{-1}})... | {
"Header 1": "**20–4.** The Second Law of Thermodynamics States That the Entropy of an Isolated System Increases as a Result of a Spontaneous Process",
"Header 2": "**21–8.** The Spectroscopic Entropies of a Few Substances Do Not Agree with the Calorimetric Entropies",
"Header 3": "**Problems**",
"token_count"... |
| T<br>/<br>K | C<br>1<br>1<br>/<br>J<br>K<br>l<br>p<br>m<br>o<br>·<br>-<br>-<br>- | /<br>T<br>K | C<br>1<br>1<br>l<br>/<br>J<br>K<br>p<br>m<br>o<br>·<br>-<br>-<br>- |
|-------------|--------------------------------------------------------------------|------------------|------------------------------------------... | {
"Header 1": "**20–4.** The Second Law of Thermodynamics States That the Entropy of an Isolated System Increases as a Result of a Spontaneous Process",
"Header 2": "**21–8.** The Spectroscopic Entropies of a Few Substances Do Not Agree with the Calorimetric Entropies",
"Header 3": "**Problems**",
"token_count"... |
Why is there a discrepancy between the calculated value and the experimental value?
$$\begin{split} \overline{C}_P[\mathrm{CO}(\mathbf{s_1})]/R &= -2.820 + (0.3317 \; \mathrm{K}^{-1})T - (6.408 \times 10^{-3} \; \mathrm{K}^{-2})T^2 \\ &\quad + (6.002 \times 10^{-5} \; \mathrm{K}^{-3})T^3 \\ &\quad 10 \; \mathrm{K} \l... | {
"Header 1": "**20–4.** The Second Law of Thermodynamics States That the Entropy of an Isolated System Increases as a Result of a Spontaneous Process",
"Header 2": "**21–8.** The Spectroscopic Entropies of a Few Substances Do Not Agree with the Calorimetric Entropies",
"Header 3": "**Problems**",
"token_count"... |
Compare your answer with the experimental value of 245.5 J.K-'·mol- 1•
- **21-34.** The vibrational and rotational constants for HF(g) within the harmonic oscillator-rigid rotator model are ii0<sup>=</sup>3959 cm- 1 and E*0*<sup>=</sup>20.56 cm- <sup>1</sup> . Calculate the standard molar entropy of HF(g) at 298.15 K. ... | {
"Header 1": "**20–4.** The Second Law of Thermodynamics States That the Entropy of an Isolated System Increases as a Result of a Spontaneous Process",
"Header 2": "**21–8.** The Spectroscopic Entropies of a Few Substances Do Not Agree with the Calorimetric Entropies",
"Header 3": "**Problems**",
"token_count"... |
Given that $T_{\text{vap}} = 337.7 \text{ K}$ , $\Delta_{\text{vap}} \overline{H}(T_{\text{b}}) = 36.5 \text{kJ} \cdot \text{mol}^{-1}$ , $\overline{C}_p[CH_3OH(I)] = 81.12 \text{J} \cdot \text{K}^{-1} \cdot \text{mol}^{-1}$ , and $\overline{C}_p[CH_3OH(g)] = 43.8 \text{ J} \cdot \text{K}^{-1} \cdot \text{mol}^{-1}... | {
"Header 1": "**20–4.** The Second Law of Thermodynamics States That the Entropy of an Isolated System Increases as a Result of a Spontaneous Process",
"Header 2": "**21–8.** The Spectroscopic Entropies of a Few Substances Do Not Agree with the Calorimetric Entropies",
"Header 3": "**Problems**",
"token_count"... |
Let's consider a system with its volume and temperature held constant. The criterion that *dS* 2:: 0 for a spontaneous process does not apply to a system at constant temperature and volume because the system is not isolated; a system must be in thermal contact with a thermal reservoir to be at constant temperature. If ... | {
"Header 1": "**20–4.** The Second Law of Thermodynamics States That the Entropy of an Isolated System Increases as a Result of a Spontaneous Process",
"Header 2": "**22-1.** The Sign of the Helmholtz Energy Change Determines the Direction of a Spontaneous Process in a System at Constant Volume and Temperature",
... |
Most reactions occur at constant pressure rather than at constant volume because they are open to the atmosphere. Let's see what the criterion of spontaneity is for a system at constant temperature and pressure. Once again, we start with Equation 22.1, but now we substitute *d S* ::=:: *8q* j *T* and *8 w* = - *P d V* ... | {
"Header 1": "**20–4.** The Second Law of Thermodynamics States That the Entropy of an Isolated System Increases as a Result of a Spontaneous Process",
"Header 2": "**22-2.** The Gibbs Energy Determines the Direction of a Spontaneous Process for a System at Constant Pressure and Temperature",
"token_count": 2045... |
If *T* < 273.15 K, then !3.fus G > 0, indicating that ice will not spontaneously melt under these conditions. If *T* > 273.15 K, then !3.fusG < 0, indicating that ice will melt under these conditions.
The value of L'l G can be related to the maximum work that can be obtained from a spontaneous process carried out at ... | {
"Header 1": "**20–4.** The Second Law of Thermodynamics States That the Entropy of an Isolated System Increases as a Result of a Spontaneous Process",
"Header 2": "**22-2.** The Gibbs Energy Determines the Direction of a Spontaneous Process for a System at Constant Pressure and Temperature",
"token_count": 1357... |
A number of the thermodynamic functions we have defined cannot be measured directly. Consequently, we need to be able to express these quantities in terms of others that can be experimentally determined. To do so, we start with the definitions of A and G, Equations 22.4 and 22.11. Differentiate Equation 22.4 to obtain ... | {
"Header 1": "**20–4.** The Second Law of Thermodynamics States That the Entropy of an Isolated System Increases as a Result of a Spontaneous Process",
"Header 2": "**22–3.** Maxwell Relations Provide Several Useful Thermodynamic Formulas",
"token_count": 1989,
"source_pdf": "datasets/websources/biochem/F814BC... |
#### FXAMPIF 22-4
In Example 20–2, we stated we would prove later that the energy of a gas that obeys the equation of state
$$P(\overline{V} - b) = RT$$
is independent of the volume. Use Equation 22.22 to prove this.
SOLUTION: For $P(\overline{V} - b) = RT$ ,
$$\left(\frac{\partial P}{\partial T}\right)_{\... | {
"Header 1": "**20–4.** The Second Law of Thermodynamics States That the Entropy of an Isolated System Increases as a Result of a Spontaneous Process",
"Header 2": "**22–3.** Maxwell Relations Provide Several Useful Thermodynamic Formulas",
"token_count": 1501,
"source_pdf": "datasets/websources/biochem/F814BC... |
Equation 22.18a can be used directly to give the volume dependence of the Helmholtz energy. By integrating at constant temperature, we have
$$\Delta A = -\int_{V_1}^{V_2} P dV \qquad \text{(constant } T\text{)}$$
(22.28)
For the case of an ideal gas, we have
$$\Delta A = -nRT \int_{V_1}^{V_2} \frac{dV}{V} = -nRT ... | {
"Header 1": "**20–4.** The Second Law of Thermodynamics States That the Entropy of an Isolated System Increases as a Result of a Spontaneous Process",
"Header 2": "**22–3.** Maxwell Relations Provide Several Useful Thermodynamic Formulas",
"Header 3": "22-4. The Enthalpy of an Ideal Gas Is Independent of Pressu... |
We may seem to be deriving a lot of equations in this chapter, but they can be organized neatly by recognizing that the energy, enthalpy, entropy, Helmholtz energy, and Gibbs energy depend upon natural sets of variables. For example, Equation 21.1 summarizes the First and Second Laws of Thermodynamics by
$$dU = TdS -... | {
"Header 1": "**20–4.** The Second Law of Thermodynamics States That the Entropy of an Isolated System Increases as a Result of a Spontaneous Process",
"Header 2": "**22–5.** The Various Thermodynamic Functions Have Natural Independent Variables",
"token_count": 1822,
"source_pdf": "datasets/websources/biochem... |
One of the most important applications of Equation 22.33 involves the correction for nonideality that we make to obtain the standard molar entropies of gases. The standard molar entropies of gases tabulated in the literature are expressed in terms of a hypothetical ideal gas at one bar and at the same temperature. This... | {
"Header 1": "**20–4.** The Second Law of Thermodynamics States That the Entropy of an Isolated System Increases as a Result of a Spontaneous Process",
"Header 2": "**22–6.** The Standard State for a Gas at Any Temperature Is the Hypothetical Ideal Gas at One Bar",
"token_count": 1503,
"source_pdf": "datasets/... |
Both of Equations 22.31 are useful because they tell us how the Gibbs energy varies with pressure and with temperature. Let's look at Equation 22.31*b* first. We can use Equation 22.31*b* to calculate the pressure dependence of the Gibbs energy:
$$\Delta G = \int_{P_1}^{P_2} V dP \qquad \text{(constant } T\text{)} \t... | {
"Header 1": "**20–4.** The Second Law of Thermodynamics States That the Entropy of an Isolated System Increases as a Result of a Spontaneous Process",
"Header 2": "**22–7.** The Gibbs–Helmholtz Equation Describes the Temperature Dependence of the Gibbs Energy",
"token_count": 2029,
"source_pdf": "datasets/web... |
Two phases in equilibrium with each other have the same value of G, so G(T) is continuous at a phase transition. Figure 22.7
**FIGURE 22.7** A plot of $\overline{G}(T) - \overline{H}(0)$ versus T for benzene. Note that $\overline{G}(T) - \overline{H}(0)$ is continuous but its deriva... | {
"Header 1": "**20–4.** The Second Law of Thermodynamics States That the Entropy of an Isolated System Increases as a Result of a Spontaneous Process",
"Header 2": "**22–7.** The Gibbs–Helmholtz Equation Describes the Temperature Dependence of the Gibbs Energy",
"token_count": 383,
"source_pdf": "datasets/webs... |
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