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Thermodynamic quantities in terms of g(r)

In the canonical ensemble, the average energy is given by

\[E = - {\partial \over \partial \beta} \ln Q (N, V, \beta )\]

\[ \ln Q (N, V, \beta ) = \ln Z_N - 3N \ln \lambda (\beta ) - \ln N ! \]

Therefore,

\[ E = {3N \over \lambda}{\partial \lambda \over \partial \beta } - {1 \over Z_N}{\partial Z_N \over \partial \beta }\]

Since

\[ \lambda = \left [ {\beta h^2 \over 2 \pi m } \right ] ^{1/2} \]

\[ {\partial \lambda \over \partial \beta } = {1 \over 2 \beta } \lambda \]

Thus,

\[ E = {3 \over 2} NkT + {1 \over Z_N} \int dr_1 \cdots dr_N U(r_1, \cdots , r_N ) e^{\beta U(r_1, \cdots , r_N)}\]

\[ = {3 \over 2} NkT + \langle U \rangle \]

In order to compute the average energy, therefore, one needs to be able to compute the average of the potential \(\langle U \rangle \). In general, this is a nontrivial task, however, let us work out the average for the case of a pairwise-additive potential of the form

\[ U (r_1, \cdots , r_N ) = {1 \over 2} \sum _{i, j, i \ne j } u(|r_i - r_j|) \equiv U_{pair} (r_1, \cdots , r_N)\]

i.e., U is a sum of terms that depend only the distance between two particles at a time. This form turns out to be an excellent approximation in many cases. U therefore contains N(N-1) total terms, and \(\langle U \rangle \) becomes

\[ \langle U \rangle = {1 \over 2Z_N} \sum _{i,j, i \ne j} \int dr_1 \cdots dr_N u (|r_i - r_j|)e^{-\beta U_{pair} (r_1, \cdots, r_N)}\]

\[ = {N (N - 1) \over 2Z_N} \int dr_1 \cdots dr_N u (|r_1 - r_2|)e^{-\beta U_{pair} (r_1, \cdots, r_N)}\]

The second line follows from the fact that all terms in the first line are the exact same integral, just with the labels changed. Thus,

\[ \langle U \rangle = {1 \over 2} \int dr_1 dr_2 u (|r_1 - r_2|) \left [ {N (N - 1) \over Z_N} \int dr_3 \cdots dr_N e^{-\beta U_{pair} (r_1, \cdots, r_N)}\right ]\]

\[ = {1 \over 2 } \int dr_1dr_2 u(|r_1 - r_2|)\rho^{(2)} (r_1, r_2 ) \]

\[ = {N^2 \over 2V^2} \int \int dr_1dr_2 u(|r_1 - r_2|)p^{(2)} (r_1, r_2 ) \]

Once again, we change variables to \(r = r_1 - r_2 \) and \(R = {(r_1 + r_2) \over 2}\). Thus, we find that

\[ \langle U \rangle = {N^2 \over 2V^2} \int drdRu (r)\tilde {g}^{(2)} (r, R) \]

\[ = {N^2 \over 2V^2} \int dru (r) \int dR\tilde {g}^{(2)} (r, R) \]

\[ = {N^2 \over 2V^2} \int dru (r) \tilde {g} (r) \]

\[ = {N^2 \over 2V^2} \int _0^{\infty} dr4 \pi r^2 u (r) g(r) \]

Therefore, the average energy becomes

\[ E = {3 \over 2} NkT + {N \over 2} 4\pi \rho \int _0^{\infty} drr^2 u(r) g(r) \]

Thus, we have an expression for E in terms of a simple integral over the pair potential form and the radial distribution function. It also makes explicit the deviation from ``ideal gas'' behavior, where E=3NkT/2.

By a similar procedure, we can develop an equation for the pressure P in terms of g(r). Recall that the pressure is given by

\[ P = {1 \over \beta } {\partial \ln Q \over \partial V} \]

\[ = {1 \over \beta Z_N}{\partial Z_N \over \partial \beta } \]

The volume dependence can be made explicit by changing variables of integration in \(Z_N \) to

\[ s_i = V^{-1/3} r_i \)

Using these variables, \(Z_N\)  becomes

\[ Z_N = V^N \int ds_1 \cdots ds_N e^{-\beta U(V^{1/R}_1, \cdots, V^{1/R}_N)} \]

Carrying out the volume derivative gives

\[ {\partial Z_N \over \partial V} = {N \over V} Z_N - \beta V^N \int ds_1 \cdots ds_N {1 \over 3V} \sum _{i=1}^N r_i \cdot {\partial U \over \partial r_i} e^{-\beta U (V^{1/R}_1, \cdots, V^{1/R}_N)} \]

\[ = {N \over V} Z_N + \beta \int dr_1 \cdots dr_N {1 \over 3V} \sum _{i=1} r_i \cdot F_i e^{-\beta U (r_1, \cdots, r_N)}\]  

Thus,

\[{1 \over Z_N}{\partial Z_N \over \partial V} = {N\over V} + {\beta \over 3V} \left < \sum _{i=1}^N r_i \cdot F_i \right > \]

 

Let us consider, once again, a pair potential. We showed in an earlier lecture that

\[ \sum _{i=1}^N r_i \cdot F_i = \sum _{i=1} \sum _{j=1, j\ne i}^N r_i \cdot F_{ij} \]

where \(F_{ij}\) is the force on particle i due to particle j. By interchaning the i and j summations in the above expression, we obtain

\[\sum _{i=1}^N r_i \cdot F_i = {1 \over 2} \left [ \sum _{i,j,i\ne j} r_i \cdot F_{ij} + \sum _{i,j,i\ne j} r_j \cdot F_{ij} \right ] \]

However, by Newton's third law, the force on particle i due to particle j is equal and opposite to the force on particle j due to particle i:

\[ F_{ij} = - F_{ji}\]

Thus,

\[ \sum_{i=1}^N r_i \cdot F_i = {1 \over 2} \left [ \sum _{i,j,i\ne j} r_i \cdot F_{ij} - \sum _{i,j,i\ne j} r_j \cdot F_{ij} \right ] = {1 \over 2} \sum _{i,j,i\ne j} (r_i - r_j ) \cdot F_{ij} \equiv {1 \over 2} \sum _{i,j,i\ne j} r_{ij} \cdot F_{ij} \]

The ensemble average of this quantity is

\[ {\beta \over 3V} \left < \sum _{i=1}^N r_i \cdot F_i \right > = {\beta \over  6V} \left < \sum _{i, j, i \ne j} r_{ij} \cdot F_{ij} \right > = {\beta \over 6VZ_N } \int dr_1 \cdots dr_N \sum _{i, j , i \ne j} r_{ij} \cdot F_{ij} e^{-\beta U_{pair} (r_1, \cdots, r_N)} \]

As before, all integrals are exactly the same, so that

\[ {\beta \over 3V} \left < \sum _{i=1}^N r_i \cdot F_i \right > = {\beta N (N - 1) \over 6V Z_N} \int d_1 \cdot r_N r_{12} \cdot F_{12} e^{-\beta U_{pair} (r_1, \cdots, r_N)} \]

\[ = {\beta \over 6V} \int dr_1 dr_2 r_{12} \cdot F_{12} \left [ {N (N - 1) \over Z_N } \int dr_3 \cdots dr_N e^{-\beta U_{pair} (r_1, \cdots, r_N)} \right ] \]

\[ = {\beta \over 6V} \int dr_1 dr_2 r_{12} \cdot F_{12} \rho^{(2)} (r_1, r_2)\]

\[ = {\beta N^2 \over 6V^3 } \int dr_1 dr_2 r_{12} \cdot F_{12}g^{(2)} (r_1, r_2) \]

Then, for a pair potential, we have

\[ F_{12} = - {\partial U_{pair} \over \partial r_{12} } = - u' (| r_1 - r_2|) {(r_1 - r_2) \over |r_1 - r_2| } = -u' (r_{12}) {r_{12} \over r_{12} } \]

where u'(r) = du/dr, and \(r_{12} = |r_{12}| \). Substituting this into the ensemble average gives

\[ {\beta \over 3V} \left < \sum _{i=1}^N r_i \cdot F_i \right > = - {\beta N^2 \over 6V^3} \int dr_1 dr_2 u' (r_{12} ) r_{12} g^{(2)} (r_1, r_2) \]

As in the case of the average energy, we change variables at this point to \(r = r_1 - r_2 \) and \(R = {(r_1 + r_2 ) \over 2} \). This gives

\[ {\beta \over 3V } \left < \sum _{i=1}^N r_i \cdot F_i \right > = - {\beta N^2 \over 6V^3} \int dr dR u^t (r) r \tilde {g} ^{(2)} (r, R)\] 

\[ = - {\beta N^2 \over 6V^2 } \int dr u^t (r) r \tilde {g} (r) \]

\[ = - {\beta N^2 \over 6V^2 } \int _0^{\infty} dr 4 \pi r^3 u^t (r) g (r) \]

Therefore, the pressure becomes

\[ {P \over kT } = \rho - {\rho ^2 \over 6kT } \int _0^{\infty} dr 4 \pi r^3 u^t (r) g(r) \]

which again gives a simple expression for the pressure in terms only of the derivative of the pair potential form and the radial distribution function. It also shows explicitly how the equation of state differs from the that of the ideal gas \({P \over kT } = \rho \).

From the definition of g(r) it can be seen that it depends on the density \(\rho \) and temperature T: \(g (r) = g (r; \rho , T ) \). Note, however, that the equation of state, derived above, has the general form

\[ {P \over kT } = \rho + B \rho ^2 \]

which looks like the first few terms in an expansion about ideal gas behavior. This suggests that it may be possible to develop a general expansion in all powers of the density \(\rho \) about ideal gas behavior. Consider representing \(g (r; \rho, T ) \) as such a power series:

\[ g (r; \rho , T) = \sum _{j=0}^{\infty} \rho ^j g_j (r; T) \]

Substituting this into the equation of state derived above, we obtain

\[ {P \over kT } = \rho + \sum _{j=0}^{\infty} B_{j+2} (T) \rho ^{j+2} \] 

This is known as the virial equation of state, and the coefficients \(B_{j+2} (T) \) are given by

\[ B_{j+2}(T) = - {1 \over 6kT} \int _0^{\infty} dr 4\pi r^3 u^t (r) g_j (r; T) \]

are known as the virial coefficients. The coefficient \(B_2 (T) \) is of particular interest, as it gives the leading order deviation from ideal gas behavior. It is known as the second virial coefficient. In the low density limit, \(g (r; \rho , T) \approx g_0 (r; T) \) and \(B_2 (T) \) is directly related to the radial distribution function.