Chapter 2 Statistical Thermodynamics
2.3 Energy Mode and Energy Levels
values of εi are held fixed during the differentiation. The result is
∂ ln Z
With similar treatments in light of Eqs. (2.26) and (2.27), the Helmholtz free energy F is determined to be
F = U− T S = −nRT ln Z. (2.28)
The equations developed above permit the evaluation of pertinent thermo-dynamic properties where Z function should be determined by methods of statistics. Z function of each energy mode depends in turn upon the allowed energy levels for that mode. Hence we need, next, to investigate techniques to obtain values of the discrete energy levels.
2.3 Energy Mode and Energy Levels [2, 5]
The total energy of a molecule can be split into energy resulting from different modes of motion. Roughly speaking, a molecular energy state is the sum of electronic, vibrational, rotational, nuclear and translational components, consequently, ε = εe + εv + εrt+ εn+ εt. In the quantum mechanics, also known as the wave mechanics, the general method of attacking a problem is to set up and solve an equation known as Schr¨odinger’s equation. For a single particle in three dimensions,
i∂
∂tΨ(r, t) = −2
2m∇2Ψ(r, t) + V (r)Ψ(r, t). (2.29) This Schr¨odinger equation for a system where r = (x, y, z) is the particle’s po-sition in a three-dimensional space,Ψ(r, t) denotes the wave function, which is the amplitude for the particle at a given position r at any given time t, m is the mass of the particle, and V (r) is the potential energy of the particle at the position r.
48 Chapter 2 Statistical Thermodynamics
In thermodynamics one is primarily concerned with particles which are restrained to certain regions of the space. In this case, time-independent standing waves are set up. Eq. (2.29) will generally yield to a separation-of-variables solution of the formΨ(r, t) = f(t)ψ(r). Thus, the equation becomes
i
. Since the left-hand side depends upon t alone and the right-hand side upon r only, each side must equal the same constant named ε. Thus,
V (r)ψ− 2
2m∇2ψ = εψ. (2.30)
Suppose that a particle translates freely in a box. The motion is free where no external force fields or intermolecular forces act upon it except on the box walls. Such simplest case would be unidimensional motion in the interval (0, L) with collisions on the “walls”. Thus, V (r) = 0 in (0, L) and infinite at x = 0 and L. Since no wave function can exist as V → ∞, ψ(0) and ψ(L) must both vanish. And since ψ function should be continuous, it should still be close to zero at x = 0 and L, where V (r) = 0. We can therefore drop V (r) from the consideration and write Schr¨odinger equation in the form
−2
Substitution of the first boundary condition of ψ(0) = 0 gives B = 0 and that of the second one of ψ(L) = 0 specifies the eigenvalues as εx= h2P
8mL2n2xwith nx= 1, 2, 3,· · ·. The term nxis known as the translational quantum number in the x direction. Similar equations are valid in the y and z directions. For a free particle in a three-dimensional cubic box of volume, V = L3, the total translational energy is
εt= h2P
8mV2/3(n2x+ n2y+ n2z) = n2ih2P
8mV2/3 (2.32)
where n2i = n2x+ n2y+ n2z, and nx, ny, nz are integers of 1, 2, 3,· · ·, etc.
The degeneracy gi of a level, or the number of energy states in the level, is calculated in terms of the translational quantum numbers. The result is
gi= 1 8
4π 3 n2i =π
6n3i. (2.33)
In order to evaluate the translational partition function Zt, the translational energy equation (2.32) is substituted into the expression for Z,
Zt=
2.3 Energy Mode and Energy Levels 49
The value of Z depends upon the summation over all available levels of energy. Before trying to reduce this to a more compact form, let us look at the quantity, the so-called “characteristic temperature for translation”, Θt ≡ h2P/8mkV2/3. For a hydrogen molecule in a 1 cm3 cubic box, Θt ≡ 1.2× 10−14K.Θtprovides an indicator of the closeness of quantum spacing.
Under normal conditions, the intervals in the above summations are very close to each other since h2P is a very small number. When Θt << T , the summation on the right-hand side of Eq. (2.34) may be evaluated as the product of three integrals of a similar form. Considering the x direction only, Zx = For a given chemical species, Ztfunction is primarily a function of V and T of the system.
All diatomic gases and a number of triatomic gases (for examples, CO, CO2, CS2, N2O) are linear molecules. The atoms of the molecule lie in a straight line. As a first approximation for these gases, it may be assumed that the interatomic distances are fixed. The assumption of a rigid rotator is valid generally at not too high T . Under these conditions, the solution of the wave equation shows the total energy of both rotational degrees of freedom of the j-th level εrt as
εrt = j(j + 1)h2P/(8π2Ir) (2.36) where Ir is the moment of inertia of the molecule. The rotational quantum numbers j are integral values with j = 0, 1, 2, 3,· · ·. The order of magnitude for the spacing of εrt ∼ 5 to 500 J·mol−1 is much larger than that for the translational mode. Excitation of the rotational levels thus would occur at much higher T than that of the translation.
In addition to the energy-level spacing, we need information on the num-ber of energy states per level. Quantum mechanics demonstrates that there are 2j + 1 of molecular quantum states which correspond approximately to the same magnitude of the rotational energy. Therefore, in rotation,
gj= 2j + 1. (2.37)
The degeneracy of a rotational level thus increases linearly with j. Both Eqs. (2.36) and (2.37) are substituted into the expression for the rotational partition function of a rigid rotator Zrt,
Zrt=
j=0
(2j + 1) exp[−j(j + 1)h2/(8π2IrkT )]. (2.38)
50 Chapter 2 Statistical Thermodynamics
Once again we wish to develop the limiting value of this sum. Let Θrt ≡ h2P/(8π2Irk) be the characteristic rotational temperature with a dimension of T , Zrt =
j=0(2j + 1) exp[−j(j + 1)Θrt/T ]. Although the energy lev-els are more widely spaced in rotation than in translation, a continuum of energy often is a reasonable assumption. The summation of Zrt with this approach becomes integration. The evaluation of the rotational partition function by integration may be justified when Θrt << T , Zrt =
∞ 8π2IrkT /h2P. This equation is valid for a heteronuclear diatomic molecule, assuming a continuum for the energy levels and behavior as a rigid rotator.
For homonuclear molecules a symmetry number ns is necessary in the de-nominator of the rotational partition. For homonuclear diatomic molecules, ns = 2, and for heteronuclear ones ns = 1. Therefore the above equation should be revised into the general form,
Zrt = 8π2IrkT /(nsh2P) = T /(nsΘrt). (2.40) When the energy levels are not closely spaced, it may be necessary to carry out the actual summation of terms in Eq. (2.38), rather than to use an integra-tion technique. This is accomplished through the Euler-Maclaurin summaintegra-tion formula. The result is This approaches Eq. (2.40) rapidly as T > Θrt. However, Θrt is generally assumed between 2 and 100 K, Eq. (2.40) is thus valid in the range of practical interest.
The vibrational motion of a molecule can often be treated as the motion of a harmonic oscillator. A unidimensional harmonic oscillator is a particle moving about an equilibrium position (x = 0) subject to a restoring force fo that is linearly dependent upon x. Thus, fo = −ksx, where ks is the
“spring constant.” The potential energy for such a particle is defined for any conservative force field, consequently fo = −dV (x)/dx. Thus, we have, in this case, the scalar relation,
V (x) =−
x
0
fodx =ksx2/2. (2.42) Equation (2.42) is substituted into Eq. (2.30) to obtain the Schr¨odinger wave equation for a one-dimensional harmonic oscillator, d2ψ
dx2 +8π2m h2
ε−
2.3 Energy Mode and Energy Levels 51
1 2ksx2
ψ = 0. The general solution of this differential equation is given by an infinite series. The allowed vibrational energy levels are quantized, and given by
εv=
υ +1
2
hPν (2.43)
where υ = 0, 1, 2, 3,· · · is the vibrational quantum number, and ν is the fundamental frequency of oscillation in s−1. ν lies in the infrared region of the electromagnetic spectrum for diatomic gases. Recall that the equation for the rotational energy indicates the ground-level energy being zero. However, in the ground level, the vibrational energy is hν/2. The quantity hν is referred to as a quantum of energy.
The energy spacing of the vibrational levels of 4000 – 40000 J·mol−1 is extremely large. According to Eq. (2.43), the vibrational levels of a harmonic oscillator are equally spaced. The quantum mechanics also shows that the degeneracy of a one-dimensional harmonic oscillator is unity. That is gvib= 1.
The vibrational partition function may now be obtained by the usual summation process. In this case, the characteristic temperature of vibration Θv is large, being in the order of 103K. One cannot pass from summation to integration at temperatures in the range of practical interest. It is unnecessary to do so, however, because in this case the partition function can be summed exactly. The multiplicity, or degeneracy factor, of the vibrational levels is unity. The partition function constructed from the energy given in Eq. (2.43) is Zv =
υ=0exp[−(υ + 1/2)hPν/(kT )] where hPν/k ≡ Θv. This equation can be expanded as Zv = exp[−Θv/(2T )][1 + exp(−Θv/T ) + exp(−2Θv/T ) +
· · · ] and summed. We recognize that the bracketed term is in the form of a binomial expansion. Thus Zv is presented as
Zv = exp[−Θv/(2T )]
1− exp (−Θv/T ). (2.44)
In some instances it is convenient to suppress the ground-level energy εv,0 and then correct for this later. Equation (2.43) may be modified as
εv− εv,0= υhPν. (2.45)
If Eq. (2.45) is used as vibrational energy εv in the partition function of vi-bration, a constant multiplicative factor, exp[−hPν/(2kT )], has likewise been suppressed. The equation for the partition function of a harmonic oscillator is then
Zv=
1− exp−hPν kT
−1
. (2.46)
52 Chapter 2 Statistical Thermodynamics