Calculation of average thermodynamic quantities

Energy

If we denote the average energy áEñ then

 

 We use the notation that áEñ = U - U(0) where U(0) is the energy at zero Kelvin. Recalling that b = 1/kT this can be rewritten as

 This can be written compactly as

 

 

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

using the notation that áEñ = U - U(0) where U(0) is the energy at zero Kelvin. The molar internal energy of a monatomic ideal gas is áEñ = 3/2RT. The heat capacity of a monatomic ideal gas is therefore Cv = 3/2R.

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 ˝RT 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. 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 Cv 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.

 

Pressure

Pressure can also be derived from the canonical partition function.

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

From thermodynamics, pressure is expressed as

so we can write

 

In a few steps we can show that the temperature can be expressed in terms 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(XY) = Yln(X). Only one term in the ln Q depends on V.

 

Taking the derivative of NlnV with respect to V gives

Substituting this into the above equation for the pressure gives P=NkT/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. NAk = R converts the constant from a "per molecule" to a "per mole" basis.

 

Entropy

We have calculated E = U – U(0), which is the internal energy referenced to the value, U(0) at absolute zero (T = 0 K).

We can now calculate the entropy, S = k ln W

 Now recalling the definition of the Boltzmann distribution

ln pi = - bei – ln q

The entropy is,

. 

The entropy can be expressed in terms of the system partition function Q

.

Helmholtz Free Energy

The Helmholtz free energy is A = U - TS. Substituting in for U and S from above we have

A = - kT ln Q.