Average Properties - From the Microscopic to the Macroscopic

Calculation of Average Energy

If we denote the average energy <E> then

Recalling that b = 1/ k_BT(click here for more details)

This can be written compactly as

The Ensemble Partition Function

We distinguish here between the partition function of the ensemble, Q and that of an individual molecule, q. Since Q represents a sum over all states accessible to the system it can written as

where the indices i,j,k, represent energy levels of different particles.

The discussion of fermions and bosons in M&S 17-7 does not really matter for the conclusions we reach in this course.

Fermion = particle of half-integer spin (s = 1/2, 3/2 …)

Boson = particle of integer spin (s = 0, 1, 2…)

Regardless of the type of particle the molecular partition function, q represents the energy levels of one individual molecule. We can rewrite the above sum as Q = q_iq_jq_k… or Q = q^N for N particles (M&S 17-6). Note that q_i means a sum over states or energy levels accessible to molecule i and q_j means the same for molecule j.

The molecular partition function

counts the energy levels accessible to molecule i only. Q counts not only the states of all of the molecules, but all of the possible combinations of occupations of those states. However, if the particles are not distinguishable then we will have counted N! states too many (N! = N(N-1)(N-2)….). This factor is exactly how many times we can swap the indices in Q(N,V,T) and get the same value (again provided that the particles are not distinguishable). If we consider 3 particles we have i,j,k j,i,k, k,i,j k,j,i j,k,i i,k,j or 6 = 3!. This is true for fermions (s=1/2 electrons) and bosons (s=0 nuclei). Thus we write the partition function as

The Molecular Partition Function

We have seen above that the internal energy of the system can be calculated using the partition function. In fact, all thermodynamic quantities can be calculated from molecular properties using the partition function. For an ideal gas of non-interacting particles only the translational partition function matters. Indeed, our focus will be on the effects of translation properties and we will prove later that translation alone defines the properties of an ideal gas. It is important to mention at this point that molecular vibrational, rotational, and even electronic states also can contribute to the molecular partition function. This discussion complements section 17-8 in M&S.

Molecular energy levels are

e = e_a^trans + e_b^vib + e_c^rot + e_d^elec

where the indices a, b, c, d run over the levels of one particular molecule. I use different indices than M&S to differentiate the statements made about one individual molecule from those above concerning the ensemble. We can write the molecular partition function as

In chemistry 431 we derive only the translational partition function explicitly. The others discussed in section 17-8 of M&S will be discussed in 433.

The Translational Partition Function

The translational partition function is the most important one for statistical thermodynamics. As we have seen, pressure is caused by translational motion, i.e., momentum exchange with the walls of a container. For this reason it is important to understand the origin of the translational partition function (M&S 18-1). Translational energy levels are so closely spaced as that they are essentially a continuous distribution. The quantum mechanical description of the energy levels is obtained from the quantum mechanical particle in a box. This will be solved in 433 (see M&S 3-5 for more information). Here we present the result used by M&S in 18-1. The energy levels are

The box is a cube of length a. The average quantum numbers will be very large for a typical molecule. This is very different than what we find for vibration and electronic levels where the quantum numbers are small (i.e. only one or a few levels are populated). Many translational levels are populated thermally.

The translational partition function is

The three summations are identical and so they can be written as the cube of one summation

The fact that the energy levels are essentially continuous and that the average quantum number is very large allows us to rewrite the sum as an integral.

The sum started at 1 and the integral at 0. This difference is not important if the average value of n is ca. 10⁹! If we have the substitution a = h²/8ma²k_BT we can rewrite the integral as

This is a Gaussian integral. The solution of Gaussian integrals is discussed in Math chapter B of M&S and on the review section of the Website. If we now plug in for a and recognize that the volume of the box is V = a³ we have

The average energy per molecule is given by

which agrees with the kinetic theory of gases and the results derived in M&S 17-3. The justification for the first step is found in Eqn. 17-20. The second step follows from the fact that ln(abc) = ln(a) + ln(b) + ln(c). McQuarrie and Simon show you that you can rewrite the logarithm as a sum and that terms that do not depend on temperature will vanish.

The microscopic picture of the translation partition function is derived from the quantum mechanical box problem. We can also examine the velocity distribution of molecules and calculate the translation partition function from that. We will obtain the same answer, of course. The classical result follows directly from the discussion in section 25-2 where it is shown that the distribution of molecular speeds follow a Gaussian distribution.

Calculation of Average Thermodynamic Quantities

We have already seen that the average energy can be calculated using the partition function. The basic approach is to sum over the probability of a state being occupied times the value of the property in a given state. In general, for an average property A we can write

A could be energy or pressure etc. p_j is the Boltzmann probability given by p_j = e^-bej/Q.

Heat Capacity

The heat capacity is a coefficient that gives the amount of energy to raise the temperature of a substance by one degree Celsius. The heat capacity can also be described as the temperature derivative of the average energy. The constant volume heat capacity is defined by

The molar internal energy of a monatomic ideal gas is

The heat capacity of a monatomic ideal gas is therefore

These derivations come from the kinetic theory of gases and also from the definition of the translational partition function. For a monatomic gas there are three degrees of freedom per atom (these are the translations along the x, y, and z direction). Each of these translations corresponds to ½RT of energy. For an ideal diatomic gas some of the energy used to heat the gas may also go into rotational and vibrational degrees of freedom. For solids there is no translation or rotation and therefore the entire contribution to the heat capacity comes from vibrations. Given their extended nature the vibrations in solids are much lower in frequency than those of gases. Therefore, while vibrations in typical diatomic gases typically contribute little to the heat capacity, the vibrational contribution to the heat capacity of solids is the largest contribution. This is discussed more in Example 17-3 in M&S. The salient point is that the partition function approaches 1 at zero temperature. Physically, this means that only one state is accessible if the temperature is nearly at absolute zero. As the temperature is increased, there are more levels of the solid accessible by thermal energy and therefore Q increases. This also means that U increases and finally that C_v increases. In the high temperature limit in an ideal solid there are 3N vibrational modes that are accessible giving rise to a contribution to the molar heat capacity of 3R. This result is also known as the law of Dulong and Petit.

Pressure

Pressure can also be derived from the partition function.

We will show later that pressure is expressed as

The average pressure is the sum of the probability times the pressure

so we can write

In a few steps we can show that the temperature can be expressed in terms of the partition function. Since the derivative is with respect to volume, the pressure depends only the translational partition function. To begin with we remind you of the definition of the partition function

The derivative of the partition function with respect to volume is

The average pressure can then be written as

Which shows that the pressure can be expressed solely terms of the partition function.

We can use this result to derive the ideal gas law. For N particles of an ideal gas where

is the translational partition function. The utility of expressing the pressure as a logarithm is clear from the fact that we can write

We have used the property of logarithms that ln(AB) = ln(A) + ln(B) and ln(X^Y) = Yln(X). Only one term in the lnQ depends on V. Taking the derivative of NlnV with respect to V gives

Substituting this into the above equation for the pressure gives P=Nk_BT/V which is the ideal gas law. Recall that Nk = nR where N is the number of molecules and n is the number of moles. R is the universal gas constant (8.314 J/mol-K) which is nothing more than k multiplied by Avagadro’s number. N_Ak = R converts the constant from a “per molecule” to a “per mole” basis.