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Thus far we have treated the vibrational motion of a diatomic molecule by means of a harmonic-oscillator model. We saw in Section 5–3, however, that the internuclear potential energy is not a simple parabola but is more like that illustrated in Figure 5.5
(cf. also Figure 13.4). The dashed line in either of these fig... | {
"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": "13-5. Overtones Are Observed in Vibrational Spectra",
"token_count": 1825,
"source_pdf": "datasets/websources/biochem/F814BC5915875384... |
In addition to undergoing rotational and vibrational transitions as a result of absorbing microwave and infrared radiation, respectively, molecules can undergo electronic transitions. The difference in energies between electronic levels is usually such that the radiation absorbed falls in the visible or ultraviolet reg... | {
"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": "13-5. Overtones Are Observed in Vibrational Spectra",
"Header 3": "**13-6.** Electronic Spectra Contain Electronic, Vibrational, and Rot... |
SOL UTI 0 N: Using Equation 13.25 with *v'* = 0, 1, and 2, we have
$$39 699.10 = \tilde{v}_{0,0}$$
$$40 786.80 = \tilde{v}_{0,0} + \tilde{v}'_e - 2\tilde{x}'_e \tilde{v}'_e$$
$$41 858.90 = \tilde{v}_{0,0} + 2\tilde{v}'_e - 6\tilde{x}'_e \tilde{v}'_e$$
By subtracting the first equation from the second and thir... | {
"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": "13-5. Overtones Are Observed in Vibrational Spectra",
"Header 3": "**13-6.** Electronic Spectra Contain Electronic, Vibrational, and Rot... |
Figure 13.10 shows two electronic potential energy curves with the vibrational states associated with each electronic energy state indicated. In each vibrational state, the harmonic-oscillator probability densities are plotted (cf. Figure 5.8). Notice that except for the ground vibrational state, the most likely intern... | {
"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": "13-5. Overtones Are Observed in Vibrational Spectra",
"Header 3": "**13-7.** The Franck-Condon Principle Predicts the Relative Intensiti... |
In this section, we will model a polyatomic molecule as a rigid network of *N* atoms. The rotational properties of a rigid body are characterized by its *principal moments of inertia*, which are defined in the following way. Choose any set of Cartesian axes with
the origin at the center of mass of the body. The momen... | {
"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": "**13–8.** The Rotational Spectrum of a Polyatomic Molecule Depends Upon the Principal Moments of Inertia of the Molecule",
"token_count"... |
Not all symmetric top molecules have dipole moments (C6H6 and XeF4 , for example), but of those that do, most have the dipole moment directed along the symmetry axis (NH3 and CH3CN, for example). The selection rules for such molecules are
$$\Delta J = 0, \pm 1 \quad \Delta K = 0 \qquad \text{for } K \neq 0$$
$$\Del... | {
"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": "**13–8.** The Rotational Spectrum of a Polyatomic Molecule Depends Upon the Principal Moments of Inertia of the Molecule",
"token_count"... |
The vibrational spectra of polyatomic molecules tum out to be easily understood in terms of the harmonic-oscillator approximation. The key point is the introduction of normal coordinates, which we discuss in this section.
**FIGURE 13.14**
Part of the J = 8 ---+ 9 transition of CF3CCH... | {
"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": "**13-9.** The Vibrations of Polyatomic Molecules Are Represented by Normal Coordinates",
"token_count": 2015,
"source_pdf": "datasets/... |
The normal modes *<sup>1</sup>* of formaldehyde (H2CO) and chloromethane (CH3Cl) are shown in Figure 13.15. A selection rule for vibrational absorption spectroscopy is that the dipole moment of the molecule must vary during the normal mode motion. When this is so, the normal mode is said to be *infrared active.* Otherw... | {
"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": "**13-9.** The Vibrations of Polyatomic Molecules Are Represented by Normal Coordinates",
"token_count": 706,
"source_pdf": "datasets/w... |
Group theory can be used to characterize the various normal coordinates belonging to any molecule. This section uses the group-theoretic ideas presented in Chapter 12. (If you have not studied Chapter 12, then you may skip this section and go on to the next section.)
The fact that the vibrational properties of a mole... | {
"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": "**13-10.** Normal Coordinates Belong to Irreducible Representations of Molecular Point Groups",
"token_count": 360,
"source_pdf": "dat... |
For the asymmetric stretch $(Q_{ss})$ , however, we have
$$\hat{c}_{v} \left( \begin{array}{c} \dot{c} \\ \dot{c} \\ \dot{c} \\ \dot{c} \\ \dot{c} \\ \dot{c} \\ \dot{c} \\ \dot{c} \\ \dot{c} \\ \dot{c} \\ \dot{c} \\ \dot{c} \\ \dot{c} \\ \dot{c} \\ \dot{c} \\ \dot{c} \\ \dot{c} \\ \dot{c} \\ \dot{c} \\ \dot{c} \\ \d... | {
"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": "**13-10.** Normal Coordinates Belong to Irreducible Representations of Molecular Point Groups",
"token_count": 2009,
"source_pdf": "da... |
| | Ê | $\hat{C}_{2}$ | $\hat{\sigma}_v$ | $\hat{\sigma}'_v$ |
|-------------------------|---|---------------|------------------|-------------------|
| $\nu_1$ | 1 | 1 | 1 | 1 |
| $v_1$ $v_2$ $v_3$ $v_4$ | 1 | 1 | 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": "**13-10.** Normal Coordinates Belong to Irreducible Representations of Molecular Point Groups",
"token_count": 1827,
"source_pdf": "da... |
Using Table 13.7 to determine the characters of the 12-dimensional reducible representation, we obtain
$$\begin{array}{c|ccccccccccccccccccccccccccccccccccc$$
Using Equation 12.23, we get
$$\Gamma_{3N} = A_1' + A_2' + 3E' + 2A_2'' + E''$$
The *D3 <sup>h</sup>*character table shows that *Tx* and *TY* jointly bel... | {
"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": "**13-10.** Normal Coordinates Belong to Irreducible Representations of Molecular Point Groups",
"token_count": 281,
"source_pdf": "dat... |
The spectroscopic selection rules determine which transitions from one state to another are possible. The very nature of transitions implies a time-dependent phenomenon, so we must use the time-dependent Schri::idinger equation (Equation 4.15)
$$\hat{H}\Psi = i\hbar \frac{\partial \Psi}{\partial t} \tag{13.39}$$
We... | {
"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": "**13-11.** Selection Rules Are Derived from Time-Dependent Perturbation Theory",
"token_count": 1914,
"source_pdf": "datasets/websourc... |
Solving Equation 13.49 for $i\hbar da_2/dt$ gives
$$i\hbar\frac{da_{_{2}}}{dt}=a_{_{1}}(t)e^{iE_{_{2}}t/\hbar}\int\psi_{_{2}}^{*}\hat{H}^{(1)}\Psi_{_{1}}d\tau+a_{_{2}}(t)e^{iE_{_{2}}t/\hbar}\int\psi_{_{2}}^{*}\hat{H}^{(1)}\Psi_{_{2}}d\tau$$
Using Equation 13.45 for $\Psi_1$ and $\Psi_2$ finally gives
$$i\hb... | {
"Header 1": "**13–3.** Vibration–Rotation Interaction Accounts for the Unequal Spacing of the Lines in the *P* and *R* Branches of a Vibration-Rotation Spectrum",
"Header 2": "**13-11.** Selection Rules Are Derived from Time-Dependent Perturbation Theory",
"token_count": 1589,
"source_pdf": "datasets/websourc... |
We can use Equation 13.54 and the properties of the spherical harmonics to derive the selection rule for a rigid rotator. Recall that the rigid-rotator wave functions are the spherical harmonics, which are developed in Section 6-2. Once again, if we assume the electric field lies along the z-axis, then the dipole trans... | {
"Header 1": "**13–3.** Vibration–Rotation Interaction Accounts for the Unequal Spacing of the Lines in the *P* and *R* Branches of a Vibration-Rotation Spectrum",
"Header 2": "**13-11.** Selection Rules Are Derived from Time-Dependent Perturbation Theory",
"Header 3": "**13-12.** The Selection Rule in the Rigid... |
If we substitute the first two terms of the expansion in Equation 13.65 into Equation 13.64, we have:
$$(\mu_{z})_{v,v'} = N_{v} N_{v'} \mu_{0} \int_{-\infty}^{\infty} H_{v'}(\alpha^{1/2}q) H_{v}(\alpha^{1/2}q) e^{-\alpha q^{2}} dq$$
$$+ N_{v} N_{v'} \left(\frac{d\mu}{dq}\right)_{0} \int_{-\infty}^{\infty} H_{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": "**13-11.** Selection Rules Are Derived from Time-Dependent Perturbation Theory",
"Header 3": "**13-12.** The Selection Rule in the Rigid... |
In the previous section, we saw that a normal coordinate will be infrared active if the dipole moment of the molecule changes as the molecule vibrates. Thus, for example, the symmetric stretch of C02 will be infrared inactive, whereas the three other modes will be infrared active. We can use the vibrational selection 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": "**13-11.** Selection Rules Are Derived from Time-Dependent Perturbation Theory",
"Header 3": "**13-14.** Group Theory Is Used to Determi... |
- **13-1.** The spacing between the lines in the microwave spectrum of H35Cl is 6.350 x 1011 Hz. Calculate the bond length of H35Cl.
- **13-2.** The microwave spectrum of 39K<sup>127</sup>1 consists of a series of lines whose spacing is almost constant at 3634 MHz. Calculate the bond length of 39K1271.
- **13-3.** 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": "**13-11.** Selection Rules Are Derived from Time-Dependent Perturbation Theory",
"Header 3": "**Problems**",
"token_count": 1972,
"s... |
| Transitions | Frequency/cm <sup>-1</sup> |
|-------------------|----------------------------|
| 0 → 1 | 3.845 40 |
| $1 \rightarrow 2$ | 7.69060 |
| $2 \rightarrow 3$ | 11.535 50 |
| $3 \rightarrow 4$ | 15.379 90 |
| $4 \righta... | {
"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": "**13-11.** Selection Rules Are Derived from Time-Dependent Perturbation Theory",
"Header 3": "**Problems**",
"token_count": 1865,
"s... |
| i<br>i<br>T<br>t<br>r<br>a<br>n<br>s<br>o<br>n<br>s | F<br>/<br>r<br>e<br>q<br>u<br>e<br>n<br>c<br>y<br>e<br>m<br>I<br>- |
|-----------------------------------------------------|--------------------------------------------------------------------|
| 0<br>1<br>+<br>-<br>- | 3<br>8<br>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": "**13-11.** Selection Rules Are Derived from Time-Dependent Perturbation Theory",
"Header 3": "**Problems**",
"token_count": 1913,
"s... |
How do they differ? Indicate some of the allowed transitions in each case.
- **13-40.** Derive Equation 13.57 from Equation 13.55.
- **13-41.** Show that the first few associated Legendre functions satisfy the recursion formula given by Equation 13.62.
- **13-42.** Calculate the ratio of the dipole transition moments 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": "**13-11.** Selection Rules Are Derived from Time-Dependent Perturbation Theory",
"Header 3": "**Problems**",
"token_count": 2026,
"s... |
Certainly one of the most important spectroscopic techniques is nuclear magnetic resonance (NMR) spectroscopy, particularly to organic chemists and biochemists. Hardly a chemical laboratory in the world does not have at least one NMR spectrometer. You may have learned about the application of NMR to the determination o... | {
"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": "**Nuclear Magnetic Resonance** Sp~(:troscopy",
"token_count": 299,
"source_pdf": "datasets/websources/biochem/F814BC5915875384820.pdf"... |
We learned in Section 8-4 that an electron has an intrinsic spin angular momentum, whose *z* components are equal to ± *h* /2, or that it has a spin of 112, with *z* components ±1/2. We defined two spin functions, a(a) and f3(a), where a is a spin variable, which satisfy the eigenvalue equations
$$\hat{S}^{2}\alpha =... | {
"Header 1": "**13–3.** Vibration–Rotation Interaction Accounts for the Unequal Spacing of the Lines in the *P* and *R* Branches of a Vibration-Rotation Spectrum",
"Header 2": "**14-1.** Nuclei Have Intrinsic Spin Angular Momenta",
"token_count": 1813,
"source_pdf": "datasets/websources/biochem/F814BC591587538... |
A magnetic dipole will tend to align itself in a magnetic field, and its potential energy will be given by (see Problem 13-49)
$$V = -\boldsymbol{\mu} \cdot \mathbf{B} \tag{14.10}$$
where **B** is the strength of the magnetic field. The quantity **B** is defined through the equation
$$\mathbf{F} = q(\mathbf{v} \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": "**14-2.** Magnetic Moments Interact with Magnetic Fields",
"token_count": 2025,
"source_pdf": "datasets/websources/biochem/F814BC59158... |
According to Equation 14.18, the *resonance frequency* of a proton (the frequency at which a spin-state transition will occur) in a magnetic field is directly proportional to the strength of the magnetic field. Thus, for a fixed magnetic field strength, we can vary the frequency of the electromagnetic radiation until a... | {
"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": "**14–3.** Proton NMR Spectrometers Operate at Frequencies Between 60 MHz and 750 MHz",
"token_count": 888,
"source_pdf": "datasets/web... |
In Section 14–3, we showed that the two spin states of a spin 1/2 nucleus such as a proton have different energies in a magnetic field, and that the frequency associated with a transition from one state to another is given by Equation 14.18, $v = \gamma B_z/2\pi$ . According to this equation, all the hydrogen nuclei i... | {
"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": "**14–3.** Proton NMR Spectrometers Operate at Frequencies Between 60 MHz and 750 MHz",
"Header 3": "14–4. The Magnetic Field Acting upon... |
Estimate the difference in chemical shifts for the hydrogen nuclei labelled aand b. What would be the separation between the two signals on a 270-MHz spectrometer?
S 0 L UTI 0 N : The *a*signal occurs at approximately 480Hz, so using Equation 14.23, we find that
$$\delta_a = \left(\frac{480~\mathrm{Hz}}{60~\mathrm{... | {
"Header 1": "**13–3.** Vibration–Rotation Interaction Accounts for the Unequal Spacing of the Lines in the *P* and *R* Branches of a Vibration-Rotation Spectrum",
"Header 2": "**14–3.** Proton NMR Spectrometers Operate at Frequencies Between 60 MHz and 750 MHz",
"Header 3": "14–4. The Magnetic Field Acting upon... |
Because the shielding of a nucleus is caused by the enhanced electronic currents set up in the molecule by the external applied magnetic field, we can expect that the degree of shielding increases with increasing electron density around the nucleus. As Equation 14.21 shows, the larger the shielding constant, the greate... | {
"Header 1": "**13–3.** Vibration–Rotation Interaction Accounts for the Unequal Spacing of the Lines in the *P* and *R* Branches of a Vibration-Rotation Spectrum",
"Header 2": "**14-5.** Chemical Shifts Depend upon the Chemical Environment of the Nucleus",
"token_count": 1488,
"source_pdf": "datasets/websource... |
There is an important feature of NMR spectra we have not discussed yet. To see this feature, let's consider 1,1,2-trichloroethane. There are two types of hydrogen atoms in this molecule. One set contains one hydrogen atom, and the other contains two structurally equivalent hydrogen atoms. Consequently, we predict that ... | {
"Header 1": "**13–3.** Vibration–Rotation Interaction Accounts for the Unequal Spacing of the Lines in the *P* and *R* Branches of a Vibration-Rotation Spectrum",
"Header 2": "**14-5.** Chemical Shifts Depend upon the Chemical Environment of the Nucleus",
"Header 3": "14-6. Spin-Spin Coupling Can Lead to Multip... |
SOLUTION: To determine Ej*<sup>0</sup>* >, we use
$$\hat{H}^{(0)}\psi_3 = E_3^{(0)}\psi_3$$
Therefore, we have
$$\begin{split} \hat{H}^{(0)} \psi_3 &= \hat{H}^{(0)} \alpha(1) \beta(2) \\ &= -\gamma \, B_0(1 - \sigma_1) \hat{I}_{z1} \alpha(1) \beta(2) - \gamma \, B_0(1 - \sigma_2) \hat{I}_{z2} \alpha(1) \beta(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": "**14-5.** Chemical Shifts Depend upon the Chemical Environment of the Nucleus",
"Header 3": "14-6. Spin-Spin Coupling Can Lead to Multip... |
| $\hat{I}_x \alpha = \frac{\hbar}{2} \beta$ | $\hat{I}_{y}\alpha = \frac{i\hbar}{2}\beta$ | $\hat{I}_z lpha = rac{\hbar}{2} lpha$ |
|--------------------------------------------|---------------------------------------------|--------------------------------------------|
| $\hat{I}_{x}eta=rac{\hbar}{2}lpha$ ... | {
"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": "**14-5.** Chemical Shifts Depend upon the Chemical Environment of the Nucleus",
"Header 3": "14-6. Spin-Spin Coupling Can Lead to Multip... |
In the previous section we showed that the first-order spectrum of an AX system leads to an NMR spectrum consisting of two doublets. Figure 14.12 shows the spectrum of dichloromethane, in which the two hydrogen atoms are chemically equivalent (an A2 system). Note that the spectrum in this case consists of just one sing... | {
"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": "**14-7.** Spin-Spin Coupling Between Chemically Equivalent Protons Is Not Observed",
"token_count": 1459,
"source_pdf": "datasets/webs... |
Evaluate £ <sup>2</sup>through first order.
S 0 L UTI 0 N : The value of £ <sup>2</sup>through first order is given by
$$\begin{split} E_2 &= E_2^{(0)} + E_2^{(1)} \\ &= \int\!\!\int d\tau_1 d\tau_2 \phi_2^* \hat{H}^{(0)} \phi_2 + \int\!\!\int d\tau_1 d\tau_2 \phi_2^* \hat{H}^{(1)} \phi_2 \end{split} \tag{1}$$
Th... | {
"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": "**14-7.** Spin-Spin Coupling Between Chemically Equivalent Protons Is Not Observed",
"Header 3": "**EXAMPLE 14-7**",
"token_count": 12... |
The splitting observed for 1,1,2-trichloroethane in Figure 14.8 shows a doublet and a triplet, and that for chloroethane in Figure 14.14 shows a triplet and a quartet. The
#### FIGURE 14.13
The energy levels of an $A_2$ system calculated by first-order perturbation theory. The two ... | {
"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": "**14-7.** Spin-Spin Coupling Between Chemically Equivalent Protons Is Not Observed",
"Header 3": "**14–8.** The n + 1 Rule Applies Only ... |
The relative simplicity of first-order spectra occurs because the spin-spin coupling constants are small relative to the separation of the multiplets. When this is the case, we can use first-order perturbation theory to calculate spectra, as we did in the previous sections. When this is not the case, we can still predi... | {
"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": "**14-9.** Second-Order Spectra Can Be Calculated Exactly Using the Variational Method",
"token_count": 1905,
"source_pdf": "datasets/w... |
These give
$$\begin{split} E_1 &= -h\nu_0 \left( 1 - \frac{\sigma_1 + \sigma_2}{2} \right) + \frac{hJ}{4} \\ E_2 &= -\frac{hJ}{4} - \frac{h}{2} [\nu_0^2 (\sigma_1 - \sigma_2)^2 + J^2]^{1/2} \\ E_3 &= -\frac{hJ}{4} + \frac{h}{2} [\nu_0^2 (\sigma_1 - \sigma_2)^2 + J^2]^{1/2} \\ E_4 &= h\nu_0 \left( 1 - \frac{\sigma_1 +... | {
"Header 1": "**13–3.** Vibration–Rotation Interaction Accounts for the Unequal Spacing of the Lines in the *P* and *R* Branches of a Vibration-Rotation Spectrum",
"Header 2": "**14-9.** Second-Order Spectra Can Be Calculated Exactly Using the Variational Method",
"token_count": 1984,
"source_pdf": "datasets/w... |
#### **EXAMPLE 14–11**
Using the results in Table 14.6, compute the spectrum of a two-spin system for $v_0=60\,\mathrm{MHz}$ and $v_0=270\,\mathrm{MHz}$ , given that $\sigma_1-\sigma_2=0.24\times10^{-6}$ and $J=8.0\,\mathrm{Hz}$ . Sketch the spectrum in each case.
SOLUTION: At 60 MHz,
$$\begin{split} \nu_... | {
"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": "**14-9.** Second-Order Spectra Can Be Calculated Exactly Using the Variational Method",
"token_count": 1697,
"source_pdf": "datasets/w... |
- 14~1. Show how Equation 14.7 reduces to Equation 14.6 for a circular orbit.
- **14-2.** What magnetic field strength must be applied for C-13 spin transitions to occur at ·-90.0MHz?
- **14-3.** What magnetic field strength must be applied for proton spin transitions to occur at 270.0 MHz?
- **14-4.** Calculate the ma... | {
"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": "**14-9.** Second-Order Spectra Can Be Calculated Exactly Using the Variational Method",
"Header 3": "**P-roblems**",
"token_count": 16... |
Start with
$$\int \alpha^* \alpha d\tau = 1 = \frac{1}{c^2} \int (\hat{I}_+ \beta)^* (\hat{I}_+ \beta) d\tau$$
Let $\hat{I}_{+} = \hat{I}_{x} + i\hat{I}_{y}$ in the second factor in the above integral and use the fact that $\hat{I}_{x}$ and $\hat{I}_{y}$ are Hermitian to get
$$\int (\hat{I}_x \hat{I}_+ \bet... | {
"Header 1": "**13–3.** Vibration–Rotation Interaction Accounts for the Unequal Spacing of the Lines in the *P* and *R* Branches of a Vibration-Rotation Spectrum",
"Header 2": "**14-9.** Second-Order Spectra Can Be Calculated Exactly Using the Variational Method",
"Header 3": "**P-roblems**",
"token_count": 20... |
In this case, the selection rule is governed by
$$P_{x} = \int d\tau_{1} d\tau_{2} \psi_{j}^{*} (\hat{I}_{x1} + \hat{I}_{x2}) \psi_{i}$$
with a similar equation for *PY.* Using the notation given by Equations 14.30, show that the only allowed transitions are for 1 ~ 2, **1** ~ 3, 2 ~ 4, and 3 ~ 4.
**14-39.** Usin... | {
"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": "**14-9.** Second-Order Spectra Can Be Calculated Exactly Using the Variational Method",
"Header 3": "**P-roblems**",
"token_count": 46... |
The word *laser* is an acronym for *light* amplification by stimulated emission of radiation. Lasers are used in a variety of devices and applications such as supermarket scanners, optical disk storage drives, compact disc players, ophthalmic and angioplastic surgery, and military targeting. Lasers have also revolution... | {
"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": "**Lasers, Laser Spectroscopy, and Photochemistry**",
"token_count": 419,
"source_pdf": "datasets/websources/biochem/F814BC591587538482... |
A molecule will not remain in an excited state indefinitely. After an excitation to an excited electronic state, a molecule invariably will relax back to its electronic ground state. Although we will consider a diatomic molecule to illustrate the mechanisms by which an electronically excited molecule can relax back to ... | {
"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": "**15–1.** Electronically Excited Molecules Can Relax by a Number of Processes",
"token_count": 2028,
"source_pdf": "datasets/websource... |
To understand how lasers work, we need to learn about the rate at which atoms and molecules undergo radiative transitions. To illustrate the concepts of radiative decay, we will focus our discussion on atoms, so that we need consider only electronic states. Molecules can be treated in a similar way, but the mathematica... | {
"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": "**15-2.** The Dynamics of Spectroscopic Transitions Between the Electronic States of Atoms Can Be Modeled by Rate Equations",
"token_cou... |
The proportionality constant
#### **FIGURE 15.5**
The spontaneous-emission process. Light of energy $h\nu_{12} = E_2 - E_1$ is emitted by an excited atom when the atom makes a transition from the electronically excited state to the ground state.
relating the rate of excited state ... | {
"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": "**15-2.** The Dynamics of Spectroscopic Transitions Between the Electronic States of Atoms Can Be Modeled by Rate Equations",
"token_cou... |
Lasers are designed to amplify light by the stimulated emission of radiation. For this amplification to occur, a photon that passes through the sample of atoms must have a greater probability of stimulating emission from an electronically excited atom than of being absorbed by an atom in its ground state. This conditio... | {
"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": "**15-2.** The Dynamics of Spectroscopic Transitions Between the Electronic States of Atoms Can Be Modeled by Rate Equations",
"Header 3"... |
The ideas presented in Section 15-3 can be generalized to multilevel systems, and we will demonstrate here that a population inversion can be achieved in a three-level system. A schematic diagram of a three-level system is shown in Figure 15.8. Each level is once again assumed to be nondegenerate and therefore represen... | {
"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": "**15-2.** The Dynamics of Spectroscopic Transitions Between the Electronic States of Atoms Can Be Modeled by Rate Equations",
"Header 3"... |
Naturally occurring ruby is unsuitable as a laser gain medium because of its strains and crystal defects, so ruby lasers use synthetic rods grown from molten mixtures of Cr20 <sup>3</sup>and Al20 <sup>3</sup> • A typical chromium doping level is about 0.05% by mass. There are many solid-state gain media, like ruby, in ... | {
"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": "**15-2.** The Dynamics of Spectroscopic Transitions Between the Electronic States of Atoms Can Be Modeled by Rate Equations",
"Header 3"... |
| G<br>i<br>d<br>i<br>a<br>n<br>m<br>e<br>u<br>m | l<br>h<br>/<br>W<br>t<br>a<br>e<br>e<br>n<br>g<br>n<br>m<br>v | O<br>t<br>t<br>u<br>p<br>u | P<br>l<br>d<br>i<br>t<br>u<br>s<br>e<br>u<br>r<br>a<br>o<br>n |
|---------------------------------... | {
"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": "**15-2.** The Dynamics of Spectroscopic Transitions Between the Electronic States of Atoms Can Be Modeled by Rate Equations",
"Header 3"... |
In 1961, the first continuous-wave laser was reported. This device, using a mixture of gaseous helium and neon as the gain medium and a direct current power supply as the pumping source, produced light in the infrared region at 1152.3 nm. In 1962, it was demonstrated that by proper choice of resonator mirrors, a He-Ne ... | {
"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": "**15-6.** The Helium-Neon Laser Is an Electrical-Discharge Pumped, Continuous-Wave, Gas-Phase Laser",
"token_count": 1860,
"source_pdf... |
| Transition | Alnm | A/106 s-<br>1 | Relative intensity |
|--------------------------------------|-------|---------------|--------------------|
| 1<br>1<br>P+ 3p<br>5s<br>S<br>I<br>0 | 730.5 | 0.48 | 30 |
| 1<br>+ 3p3<br>5s<br>P1<br>P1 | 640.1 | 0.60 ... | {
"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": "**15-6.** The Helium-Neon Laser Is an Electrical-Discharge Pumped, Continuous-Wave, Gas-Phase Laser",
"token_count": 639,
"source_pdf"... |
Figure 15.14 shows part of the absorption spectrum of IC1(g) measured by a conventional absorption spectrometer. The displayed spectrum consists of two absorption lines in the vicinity of 17 299 cm- <sup>1</sup> • The separation between the absorption lines is approximately 0.2 cm- <sup>1</sup> • These two absorption l... | {
"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": "**15-6.** The Helium-Neon Laser Is an Electrical-Discharge Pumped, Continuous-Wave, Gas-Phase Laser",
"Header 3": "**15-7.** High-Resolu... |
One application of time-resolved laser spectroscopy is to study the dynamics of chemical reactions initiated by the absorption of light. Chemical reactions that result from
the absorption of light are called *photochemical reactions*. The following equations illustrate some of the many types of photochemical reaction... | {
"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": "**15–8.** Pulsed Lasers Can Be Used to Measure the Dynamics of Photochemical Processes",
"token_count": 1860,
"source_pdf": "datasets/... |
S 0 L U T I 0 N : The radiant energy of a 306 nm photon is
$$Q_{p} = \frac{hc}{\lambda} = \frac{(6.626 \times 10^{-34} \text{ J} \cdot \text{s})(2.998 \times 10^{8} \text{ m} \cdot \text{s}^{-1})}{306 \times 10^{-9} \text{ m}}$$
$$= 6.49 \times 10^{-19} \text{ J}$$
Therefore, the number of photons in a $1.55 \ti... | {
"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": "**15–8.** Pulsed Lasers Can Be Used to Measure the Dynamics of Photochemical Processes",
"token_count": 1439,
"source_pdf": "datasets/... |
- **15-1.** The ground-state term symbol for Oi is <sup>2</sup> IT<sup>8</sup> • The first electronic excited state has an energy of 38 795 cm- <sup>1</sup>above that of the ground state and has a term symbol of <sup>2</sup> TI". Is the radiative <sup>2</sup> IT" --+ <sup>2</sup>IT8 decay of the Oi molecule an example ... | {
"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": 1950,
"source_pdf": "datasets/websources/biochem/F814BC5915875384820.pdf"
} |
Why does your answer not come out to be 3391.3 nm? (See Example 8–10.)
- **15-17.** Using the method explained in Section 8–9, show that the states associated with a $2p^5ns$ electron configuration are ${}^3P_2$ , ${}^3P_1$ , ${}^3P_0$ , and ${}^1P_1$ .
- **15-18.** Consider the excited-state electron configura... | {
"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": 1961,
"source_pdf": "datasets/websources/biochem/F814BC5915875384820.pdf"
} |
Using your results, do you think that the individual lines between 17 299.45 and 17 299.55 em\_, in Figure 15.15 can be attributed to transitions to different excited rotational states from the *X* ( *v"* = 0, *1"* = 2) ground state?
- **15-28.** Hydrogen iodide decomposes to hydrogen and iodine when it is irradiated... | {
"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": 1986,
"source_pdf": "datasets/websources/biochem/F814BC5915875384820.pdf"
} |
**15-40.** The Beer-Lambert law (Problem 15-38) can also be written in terms of the natural logarithm instead ,of the base ten logarithm:
$$A_e = \ln \frac{I_0}{I} = \kappa c I$$
In this form, the constant *K* is called the *molar napierian absorption coefficient,* and *Ae* is called the *napierian absorbance.* W... | {
"Header 1": "**13–3.** Vibration–Rotation Interaction Accounts for the Unequal Spacing of the Lines in the *P* and *R* Branches of a Vibration-Rotation Spectrum",
"Header 2": "**Problems**",
"token_count": 464,
"source_pdf": "datasets/websources/biochem/F814BC5915875384820.pdf"
} |
You learned in high school that a quadratic equation *ax <sup>2</sup>*+ *bx* + *<sup>c</sup>*= 0 has two roots, given by the so-called quadratic formula:
$$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$
Thus, the two values of *x* (called roots) that satisfy the equation *<sup>x</sup> <sup>2</sup>*+ *3x* - 2 = 0 are
$$... | {
"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": "**NUMERICAL METHODS**",
"token_count": 2019,
"source_pdf": "datasets/websources/biochem/F814BC5915875384820.pdf"
} |
Recall a fundamental theorem of calculus, which says that if
$$F(x) = \int_{a}^{x} f(u)du$$
FIGURE G.3 The integral of f(u) from a to b is given by the shaded area.
then
$$\frac{dF}{dx} = f(x)$$
The function *F(x)* is sometimes called the antiderivative of *f(x).* If there is no... | {
"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": "**NUMERICAL METHODS**",
"token_count": 2005,
"source_pdf": "datasets/websources/biochem/F814BC5915875384820.pdf"
} |
SOLUTION: At T = 103 K, the basic integral to evaluate numerically is
$$I = \int_0^3 \frac{x^4 e^x}{(e^x - 1)^2} \, dx$$
Using the trapezoidal approximation (Equation G.5) and Simpson's rule (Equation G.6), we find the following values of I:
| n | h | $I_n$ (trapezoidal) | $I_{2n}$ (Simpson's rule) |
|--... | {
"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": "**NUMERICAL METHODS**",
"token_count": 395,
"source_pdf": "datasets/websources/biochem/F814BC5915875384820.pdf"
} |
- **G-1.** Solve the equation *<sup>x</sup> <sup>5</sup>*+ *2x <sup>4</sup>*+ *4x* = 5 to four significant figures for the root that lies between 0 and 1.
- **G-2.** Use the Newton-Raphson method to derive the iterative formula
$$x_{n+1} = \frac{1}{2} \left( x_n + \frac{A}{x_n} \right)$$
for the value of .;:4. This... | {
"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": 1621,
"source_pdf": "datasets/websources/biochem/F814BC5915875384820.pdf"
} |
To this point, we have learned about the properties of individual atoms and molecules. For most of the rest of the book, we will study systems consisting of large numbers of atoms and molecules. In particular, we will explore the relations between the macroscopic properties of systems and the dependence these propertie... | {
"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 Properties of Gases**",
"token_count": 216,
"source_pdf": "datasets/websources/biochem/F814BC5915875384820.pdf"
} |
If a gas is sufficiently dilute that its constituent molecules are so far apart from each other on the average that we can ignore their interactions, it obeys the equation of state
$$PV = nRT (16.1a)$$
If we divide both sides of this equation by *n,* we obtain
$$P\overline{V} = RT \tag{16.1b}$$
where V = V / *n... | {
"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": "**16-1.** All Gases Behave Ideally If They Are Sufficiently D i I ute",
"token_count": 1852,
"source_pdf": "datasets/websources/bioche... |
1 bar = 10^5 \text{ Pa} = 0.1 \text{ MPa}
```
been achieved in the laboratory. The temperature of absolute zero (0 K) corresponds to a substance that has no thermal energy. There is no fundamental limit to the maximum value of T. There are, of course, practical limitations, and the highest value of T achieved in 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": "**16-1.** All Gases Behave Ideally If They Are Sufficiently D i I ute",
"token_count": 1594,
"source_pdf": "datasets/websources/bioche... |
The ideal-gas equation is valid for all gases at sufficiently low pressures. As the pressure on a given quantity of gas is increased, however, deviations from the ideal-gas equation appear. These deviations can be displayed graphically by plotting *P VIR T* as a function of pressure, as shown in Figure 16.3. The quanti... | {
"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": "**16-2.** The van der Waals Equation and the Redlich-Kwong Equation Are Examples of Two-Parameter Equations of State",
"token_count": 20... |
#### **EXAMPLE 16-2**
Use the van der Waals equation to calculate the molar volume of ethane at 300 K and 200 atm.
S 0 L UTI 0 N : When we try to solve the van der Waals equation for V, we obtain a cubic equation,
$$\overline{V}^3 - \left(b + \frac{RT}{P}\right)\overline{V}^2 + \frac{a}{P}\overline{V} - \frac{a... | {
"Header 1": "**13–3.** Vibration–Rotation Interaction Accounts for the Unequal Spacing of the Lines in the *P* and *R* Branches of a Vibration-Rotation Spectrum",
"Header 2": "**16-2.** The van der Waals Equation and the Redlich-Kwong Equation Are Examples of Two-Parameter Equations of State",
"token_count": 19... |
SOLUTION: Substitute $T=300\,\mathrm{K}$ , $P=200\,\mathrm{atm}$ , $A=97.539\,\mathrm{dm}^6\cdot\mathrm{atm}\cdot\mathrm{mol}^{-1}\cdot\mathrm{K}^{1/2}$ , and $B=0.045153\,\mathrm{dm}^3\cdot\mathrm{mol}^{-1}$ into Equation 16.9, to obtain
$$\overline{V}^3 - 0.1231\overline{V}^2 + 0.02056\overline{V} - 0.001271... | {
"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": "**16-2.** The van der Waals Equation and the Redlich-Kwong Equation Are Examples of Two-Parameter Equations of State",
"token_count": 86... |
A remarkable feature of equations of state that can be written as cubic equations in *V* is that they describe both the gaseous *and* the liquid regions of a substance. To understand this feature, we start by discussing some experimentally determined plots of P as a function of V at constant T, which are commonly calle... | {
"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": "**16-3.** A Cubic Equation of State Can Describe Both the Gaseous and Liquid States",
"token_count": 2034,
"source_pdf": "datasets/web... |
ForT > I;\_, only one of these roots is real (the other two are complex), and forT < I;\_ and P ~ ~.all three roots are real. At T = I;\_, these three roots merge into one, and so we can write Equation 16.10 as ( *V* - *V/* = 0, or
$$\overline{V}^3 - 3\overline{V}_c\overline{V}^2 + 3\overline{V}_c^2\overline{V} - \ov... | {
"Header 1": "**13–3.** Vibration–Rotation Interaction Accounts for the Unequal Spacing of the Lines in the *P* and *R* Branches of a Vibration-Rotation Spectrum",
"Header 2": "**16-3.** A Cubic Equation of State Can Describe Both the Gaseous and Liquid States",
"token_count": 1024,
"source_pdf": "datasets/web... |
| S<br>i<br>p<br>e<br>c<br>e<br>s | T<br>J<br>K | P<br>l<br>b<br>a<br>r<br>c | P<br>J<br>t<br>a<br>m | V<br>1<br>l<br>I<br>L<br>m<br>o<br>·<br>-<br>c | V<br>/<br>R<br>T<br>c<br>;,<br>~ |
|-------------------------------------------------------------------------|-... | {
"Header 1": "**13–3.** Vibration–Rotation Interaction Accounts for the Unequal Spacing of the Lines in the *P* and *R* Branches of a Vibration-Rotation Spectrum",
"Header 2": "**16-3.** A Cubic Equation of State Can Describe Both the Gaseous and Liquid States",
"token_count": 2150,
"source_pdf": "datasets/web... |
The experimental values of *?, VJ* R~ given in Table 16.5 show that neither equation of state is quantitative. The corresponding value for *?, VJ R* <sup>~</sup> for the Peng-Robinson equation is 0.30740 (Problem 16-28), which is closer to the experimental values than either of the values given by the van der Waals equ... | {
"Header 1": "**13–3.** Vibration–Rotation Interaction Accounts for the Unequal Spacing of the Lines in the *P* and *R* Branches of a Vibration-Rotation Spectrum",
"Header 2": "**16-3.** A Cubic Equation of State Can Describe Both the Gaseous and Liquid States",
"token_count": 1163,
"source_pdf": "datasets/web... |
Let's start with the van der Waals equation, which we can write in an interesting and practical form by substituting the second of Equations 16.12 for a and Equation 16.13a for b into Equation 16.5:
$$\left(P + \frac{3P_{\rm c}\overline{V}_{\rm c}^2}{\overline{V}^2}\right)\left(\overline{V} - \frac{1}{3}\overline{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": "**16–4.** The van der Waals Equation and the Redlich–Kwong Equation Obey the Law of Corresponding States",
"token_count": 1718,
"sourc... |
To demonstrate this point, we start with Equation 16.6 and substitute the second of Equations 16.12 for a and Equation 16.13b for b to get
$$Z = \frac{P\overline{V}}{RT} = \frac{\overline{V}}{\overline{V} - \frac{1}{3}\overline{V}_c} - \frac{3P_c\overline{V}_c^2}{RT\overline{V}}$$
Now use Equation 16.16 for $P_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": "**16–4.** The van der Waals Equation and the Redlich–Kwong Equation Obey the Law of Corresponding States",
"token_count": 1003,
"sourc... |
The most fundamental equation of state, in the sense that it has the most sound theoretical foundation, is the *virial equation of state.* The virial equation of state expresses the compressibility factor as a polynomial in 1 *IV:*
$$Z = \frac{P\overline{V}}{RT} = 1 + \frac{B_{2V}(T)}{\overline{V}} + \frac{B_{3V}(T)}... | {
"Header 1": "**13–3.** Vibration–Rotation Interaction Accounts for the Unequal Spacing of the Lines in the *P* and *R* Branches of a Vibration-Rotation Spectrum",
"Header 2": "**16–4.** The van der Waals Equation and the Redlich–Kwong Equation Obey the Law of Corresponding States",
"Header 3": "**16-5.** Second... |
An intermolecular potential that embodies the long-range (attractive) behavior of Equation 16.26 and the short-range (repulsive) behavior of Equation 16.27 is simply the sum of the two. If we taken to be 12, then
$$u(r) = \frac{c_{12}}{r^{12}} - \frac{c_6}{r^6} \tag{16.28}$$
Equation 16.28 is usually written in t... | {
"Header 1": "**13–3.** Vibration–Rotation Interaction Accounts for the Unequal Spacing of the Lines in the *P* and *R* Branches of a Vibration-Rotation Spectrum",
"Header 2": "**16–4.** The van der Waals Equation and the Redlich–Kwong Equation Obey the Law of Corresponding States",
"Header 3": "**16-5.** Second... |
We first define a reduced temperature T\* by T\* = kB TIE and let rIa = <sup>x</sup>to get
$$B_{2V}(T^*) = -2\pi\sigma^3 N_{\rm A} \int_0^\infty \left[ \exp\left\{ -\frac{4}{T^*} (x^{-12} - x^{-6}) \right\} - 1 \right] x^2 dx$$
We then divide both sides by 2rr:a*<sup>3</sup>* NAI3 to get
$$B_{2V}^*(T^*) = -3\int_... | {
"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": "**16–4.** The van der Waals Equation and the Redlich–Kwong Equation Obey the Law of Corresponding States",
"Header 3": "**16-5.** Second... |
In the previous section, we used the Leonard-Jones potential (Equation 16.29) to represent the intermolecular potential between molecules. The r- 12 term accounts for the repulsion at short distances, and the *r-<sup>6</sup>*term accounts for the attraction at larger distances. The actual form of the repulsive term is ... | {
"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": "**16–4.** The van der Waals Equation and the Redlich–Kwong Equation Obey the Law of Corresponding States",
"Header 3": "**16-6.** London... |
| Species | ~110- ° C·m | (a/4ns0)j10-30 m3 | I ;w-'s J |
|---------|-------------|-------------------|-----------|
| He | 0 | 0.21 | 3.939 |
| Ne | 0 | 0.39 | 3.454 |
| Ar | 0 | 1.63 | 2.525 |
| Kr | 0 | 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": "**16–4.** The van der Waals Equation and the Redlich–Kwong Equation Obey the Law of Corresponding States",
"Header 3": "**16-6.** London... |
Although the Lennard-Janes potential is fairly realistic, it is also difficult to use. For example, the second virial coefficient (Example 16-10) must be evaluated numerically and one must resort to numerical tables to calculate the properties of gases. Consequently, intermolecular potentials that can be evaluated anal... | {
"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": "**16–4.** The van der Waals Equation and the Redlich–Kwong Equation Obey the Law of Corresponding States",
"Header 3": "**16-7.** The va... |
The intermolecular potential that we will use is a hybrid of the hard-sphere potential and the Lennard-Jones potential
$$u(r) = -\frac{c_6}{r^6} \qquad r < \sigma \tag{16.44}$$
We substitute this potential into Equation 16.25 to obtain
$$B_{2V}(T) = -2\pi N_{\rm A} \int_0^{\sigma} (-1)r^2 dr - 2\pi N_{\rm A} \int... | {
"Header 1": "**13–3.** Vibration–Rotation Interaction Accounts for the Unequal Spacing of the Lines in the *P* and *R* Branches of a Vibration-Rotation Spectrum",
"Header 2": "**16–4.** The van der Waals Equation and the Redlich–Kwong Equation Obey the Law of Corresponding States",
"Header 3": "**16-7.** The va... |
- **16-1.** In an issue of the journal *Science* a few years ago, a research group discussed experiments in which they determined the structure of cesium iodide crystals at a pressure of 302 gigapascals (GPa). How many atmospheres and bars is this pressure?
- **16-2.** In meteorology, pressures are expressed in units o... | {
"Header 1": "**13–3.** Vibration–Rotation Interaction Accounts for the Unequal Spacing of the Lines in the *P* and *R* Branches of a Vibration-Rotation Spectrum",
"Header 2": "**16–4.** The van der Waals Equation and the Redlich–Kwong Equation Obey the Law of Corresponding States",
"Header 3": "**Problems**",
... |
**16-22.** Show that the van der Waals equation for argon at T = 142.69 K and P = 35.00 atm can be written as
$$\overline{V}^3 - 0.3664 \ \overline{V}^2 + 0.03802 \ \overline{V} - 0.001210 = 0$$
where, for convenience, we have supressed the units in the coefficients. Use the Newton-Raphson method (MathChapter 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": "**16–4.** The van der Waals Equation and the Redlich–Kwong Equation Obey the Law of Corresponding States",
"Header 3": "**Problems**",
... |
| | Ethane ( T = 500 K) | | Argon (T = 247 K) |
|-------|---------------------|-------|-------------------|
| Pfbar | 1<br>V/L·mol- | P/atm | 1<br>V/L·mol- |
| 0.500 | 83.076 | 0.500 | 40.506 |
| 2.00 | 20.723 | 2.00 | 10.106 |
| 10.00 | 4.105 ... | {
"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": "**16–4.** The van der Waals Equation and the Redlich–Kwong Equation Obey the Law of Corresponding States",
"Header 3": "**Problems**",
... |
| Pjatm | pjmol-L-<br>1 | Pjatm | pjmol·L-<br>1 |
|---------|---------------|--------|---------------|
| 0.01000 | 0.000406200 | 0.4000 | 0.0162535 |
| 0.02000 | 0.000812500 | 0.6000 | 0.0243833 |
| 0.04000 | 0.00162500 | 0.8000 | 0.0325150 |
| 0.06000 | 0.00243750 | 1.000 | 0.0406487 |
... | {
"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": "**16–4.** The van der Waals Equation and the Redlich–Kwong Equation Obey the Law of Corresponding States",
"Header 3": "**Problems**",
... |
- **16-50.** Use the critical temperatures and the critical molar volumes of argon, krypton, and xenon to illustrate the law of corresponding states with the data given in Problem 16--49.
- **16-51.** Evaluate B;v(T\*) in Equation 16.31 numerically from T\* = 1.00 to 10.0 using a packaged numerical integration progra... | {
"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": "**16–4.** The van der Waals Equation and the Redlich–Kwong Equation Obey the Law of Corresponding States",
"Header 3": "**Problems**",
... |
Recall from your course in calculus that the derivative of a function y(x) at some point *<sup>x</sup>*is defined as
$$\frac{dy}{dx} = \lim_{\Delta x \to 0} \frac{y(x + \Delta x) - y(x)}{\Delta x} \tag{H.1}$$
Physically, dy / dx expresses the variation of y when x is varied. Much of your calculus course was spent i... | {
"Header 1": "**13–3.** Vibration–Rotation Interaction Accounts for the Unequal Spacing of the Lines in the *P* and *R* Branches of a Vibration-Rotation Spectrum",
"Header 2": "**PARTIAL DIFFERENTIATION**",
"token_count": 1956,
"source_pdf": "datasets/websources/biochem/F814BC5915875384820.pdf"
} |
V to get
$$\Delta P = \left[ \frac{P(T + \Delta T, \overline{V} + \Delta \overline{V}) - P(T, \overline{V} + \Delta \overline{V})}{\Delta T} \right] \Delta T$$
$$+ \left[ \frac{P(T, \overline{V} + \Delta \overline{V}) - P(T, \overline{V})}{\Delta \overline{V}} \right] \Delta \overline{V}$$
Now let f').T--+ 0 and!')... | {
"Header 1": "**13–3.** Vibration–Rotation Interaction Accounts for the Unequal Spacing of the Lines in the *P* and *R* Branches of a Vibration-Rotation Spectrum",
"Header 2": "**PARTIAL DIFFERENTIATION**",
"token_count": 1960,
"source_pdf": "datasets/websources/biochem/F814BC5915875384820.pdf"
} |
If we apply this requirement to Equation H.l3, we find that
$$\frac{\partial}{\partial T} \left[ \frac{RT}{(\overline{V} - b)^2} - \frac{a}{T\overline{V}^2} \right] = \frac{R}{(\overline{V} - b)^2} + \frac{a}{T^2\overline{V}^2}$$
and
$$\frac{\partial}{\partial \overline{V}} \left( \frac{RT}{\overline{V} - b} \rig... | {
"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": "**PARTIAL DIFFERENTIATION**",
"token_count": 897,
"source_pdf": "datasets/websources/biochem/F814BC5915875384820.pdf"
} |
**H-1.** The isothermal compressibility, $\kappa_T$ , of a substance is defined as
$$\kappa_T = -\frac{1}{V} \left( \frac{\partial V}{\partial P} \right)_T$$
Obtain an expression for the isothermal compressibility of an ideal gas.
**H-2.** The coefficient of thermal expansion, $\alpha$ , of a substance is defin... | {
"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": 1677,
"source_pdf": "datasets/websources/biochem/F814BC5915875384820.pdf"
} |
In previous chapters, we learned that the energy states of atoms and molecules, and for all systems in fact, are quantized. These allowed energy states are found by solving the Schrodinger equation. A practical question that arises is how the molecules are distributed over these energy states at a given temperature. Fo... | {
"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 Boltzmann Factor and Partition Functions**",
"token_count": 420,
"source_pdf": "datasets/websources/biochem/F814BC5915875384820.... |
Consider some macroscopic system such as a liter of gas, a liter of water, or a kilogram of some solid. From a mechanical point of view, such a system can be described by specifying the number of particles, N, the volume, V, and the forces between the particles. Even though the system contains on the order of Avogadro'... | {
"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": "**17–1.** The Boltzmann Factor Is One of the Most Important Quantities in the Physical Sciences",
"token_count": 1913,
"source_pdf": "... |
In the limit of large $\mathcal{A}$ , which we are certainly able to take because we can make our ensemble as large as we want, $a_j/\mathcal{A}$ becomes a probability (MathChapter B), so Equation 17.12 can be written as
$$p_{j} = \frac{e^{-\beta E_{j}}}{\sum_{i} e^{-\beta E_{i}}}$$
(17.13)
where $p_j$ is 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": "**17–1.** The Boltzmann Factor Is One of the Most Important Quantities in the Physical Sciences",
"token_count": 641,
"source_pdf": "d... |
Using Equation 17.15, we can calculate the average energy of a system in an ensemble of systems. If we denote the average energy by $\langle E \rangle$ , then (see MathChapter B)
$$\langle E \rangle = \sum_{j} p_{j}(N, V, \beta) E_{j}(N, V) = \sum_{j} \frac{E_{j}(N, V) e^{-\beta E_{j}(N, V)}}{Q(N, V, \beta)}$$
(17.1... | {
"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": "17–3. We Postulate That the Average Ensemble Energy Is Equal to the Observed Energy of a System",
"token_count": 1955,
"source_pdf": "... |
(We indicate a molar quantity by an overbar.)
#### EXAMPLE 17-2
We will learn in the next chapter that for the rigid rotator-harmonic oscillator model of an ideal diatomic gas, the partition function is given by
$$Q(N, V, \beta) = \frac{[q(V, \beta)]^N}{N!}$$
where
$$q(V,\beta) = \left(\frac{2\pi m}{h^2 \beta... | {
"Header 1": "**13–3.** Vibration–Rotation Interaction Accounts for the Unequal Spacing of the Lines in the *P* and *R* Branches of a Vibration-Rotation Spectrum",
"Header 2": "17–3. We Postulate That the Average Ensemble Energy Is Equal to the Observed Energy of a System",
"token_count": 855,
"source_pdf": "d... |
The constant-volume heat capacity, *Cv,* of a system is defined as
$$C_{V} = \left(\frac{\partial \langle E \rangle}{\partial T}\right)_{N,V} = \left(\frac{\partial U}{\partial T}\right)_{N,V} \tag{17.25}$$
The heat capacity C *v* is then a measure of how the energy of the system changes with temperature at constan... | {
"Header 1": "**13–3.** Vibration–Rotation Interaction Accounts for the Unequal Spacing of the Lines in the *P* and *R* Branches of a Vibration-Rotation Spectrum",
"Header 2": "**17-4.** The Heat Capacity at Constant Volume Is the Temperature Derivative of the Average Energy",
"token_count": 1759,
"source_pdf"... |
We will show in Section 19-6 that the pressure of a macroscopic system is given by
$$P_{j}(N, V) = -\left(\frac{\partial E_{j}}{\partial V}\right)_{N} \tag{17.30}$$
Using the fact that the average pressure is given by
$$\langle P \rangle = \sum_{i} p_{j}(N, V, \beta) P_{j}(N, V)$$
we can write
$$\langle P \ra... | {
"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": "17–5. We Can Express the Pressure in Terms of a Partition Function",
"token_count": 1364,
"source_pdf": "datasets/websources/biochem/F... |
The general results we have derived up to now are valid for arbitrary systems. To apply these equations, we need to have the set of eigenvalues {E.(N, V)} for theN- <sup>J</sup> body SchrOdinger equation. In general, this is an impossible task. For many important physical systems, however, writing the total energy of 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": "17–5. We Can Express the Pressure in Terms of a Partition Function",
"Header 3": "**17-6.** The Partition Function of a System of Indepe... |
SOLUTION: For this system
$$Q(2, V, T) = \sum_{i,j=1}^{4} e^{-\beta(\varepsilon_i + \varepsilon_j)}$$
Of the 16 terms that would occur in an unrestricted evaluation of Q, only six are allowed for two identical fermions; these are the terms with energies
$$\begin{array}{ll} \varepsilon_1 + \varepsilon_2 & \qquad... | {
"Header 1": "**13–3.** Vibration–Rotation Interaction Accounts for the Unequal Spacing of the Lines in the *P* and *R* Branches of a Vibration-Rotation Spectrum",
"Header 2": "17–5. We Can Express the Pressure in Terms of a Partition Function",
"Header 3": "**17-6.** The Partition Function of a System of Indepe... |
| System | T/K | $\frac{N}{V} \left( \frac{h^2}{8mk_{\rm B}T} \right)^{3/2}$ |
|--------------------------|-----|-------------------------------------------------------------|
| Liquid helium | 4 | 1.5 |
| Gaseous helium ... | {
"Header 1": "**13–3.** Vibration–Rotation Interaction Accounts for the Unequal Spacing of the Lines in the *P* and *R* Branches of a Vibration-Rotation Spectrum",
"Header 2": "17–5. We Can Express the Pressure in Terms of a Partition Function",
"Header 3": "**17-6.** The Partition Function of a System of Indepe... |
In this section, we will explore the similarity between a system partition function, Equation 17.14, and a molecular partition function, Equation 17.39. We will start by substituting Equation 17.38 into Equation 17.21:
$$\begin{split} \langle E \rangle &= k_{\rm B} T^2 \left( \frac{\partial \ln Q}{\partial T} \right)... | {
"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": 2044,
... |
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