The Partition Function Statistical Thermodynamics. NC State University

Transcription

1 Chemistry 431 Lecture 4 The Partition Function Statistical Thermodynamics NC State University

2 Molecular Partition Functions In general, g j is the degeneracy, ε j is the energy: = j q g e βε j We assume that the energy of the lowest energy level, the ground state is ε 0 = 0. Recall that β = 1/kT Examples: A.)Two level system. B.)Infinite energy ladder. j

3 Two Level System Assume that degeneracy g 0 =g 1 =1 (i.e. single state is found at each level). q = 1+e βε Note that as T 0, q 1 and as T,, q 2. The ratio of the population in the two is states e βε where ε is the energy difference between the two states. ε

4 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 Q(N,V,T) = Σ e β(ε i + ε j + ε k +...) i,j,k,... where the indices i,j,k, represent energy levels l of different particles.

5 The molecular partition function, q represents the energy levels of one individual molecule. We can rewrite the above sum as Q = q i q j q k or Q = q N for N particles. Note that q i means a sum over states or energy levels accessible to molecule iand q j means the same for molecule j. Σ i q(v,t) = e βε i The molecular partition function counts the energy levels accessible to molecule i only.

6 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) VT) 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!. Thus we write the partition function as Q = q N Q = q N N! for distinguishable particles for indistinguishable particles

7 Translational Partition Function The translational partition function is the most important one for statistical thermodynamics. 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. 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.

8 Particle in a box in a energy levels The energy levels are ε h nx,n y,n z = n n n 8ma 2 x +n y +n z n x,n y,n z =1, 2,... 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). Manytranslational levels arepopulated thermally.

9 The translational partition function is ε e ε n x,n y,n /kt Σ z n x,n y,n z =1 q trans = e Σ Σ Σ = exp 8 2 n 2 2 +n y2 +n n 8ma2 kt x z x =1 n y =1 n z =1 The three summations are identical and so they can be written as the cube of one summation Σ h 2 q trans = exp n = 1 h 2 n 2 8ma 2 kt 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. q trans = exp 0 3 h 2 n 2 8ma 2 kt dn 3

10 The translational partition function is proportional to volume The sum started at 1 and the integral at 0. This difference is not important if the average value of n is ca. 10 9! If we have the substitution a = h 2 /8ma 2 kt we can rewrite the integral as q trans = 0 e αn2 dn 3 = π 4α 1/2 This is a Gaussian integral. The solution of Gaussian integrals is discussed the math section of the Website. If we now plug in for a and recognize that the volume of the box is V = a 3 we have 2πmkT 3/2 V q trans = 2πmkT h 2 3

11 Probability in the ensemble The ensemble partition function is: Q = e E i /kt Σ i =0 Where the ensemble energy is E J. The population of a particular state J with energy E J is given by p J = Σ J =0 e E J / kt E J / kt e E J / kt = e EJ / kt Q This known as the Boltzmann distribution. The normalization constant of the above probability is 1/Q. The sum of all of the probabilities must equal 1.

12 Calculation of average properties The importance of the canonical ensemble is evident once we begin to calculate average thermodynamic quantities. 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 M we can write < M > = Σ P M j j j M could be energy or pressure etc. P j is the Boltzmann probability given by P j = e βε j /Q.

13 Average energy If we denote the average energy <E> then E = Σ P J E = J J =0 ΣJ =0 Σ J = 0 This can be written compactly as E = ln Q E β E J e βe J e βe J

14 Consistency check with kinetic theory of gases The average energy per molecule is given by ε trans = lnqtrans β V = kt 2 lnq trans T V = kt 2 T 3 2 lnt + terms independentof T V = 3 2 kt2 1 T T T = 3 2 kt which agrees with the kinetic theory of gases. The second step follows from the fact that ln(abc) = ln(a) + ln(b) + ln(c). We can rewrite rite the logarithm as a sum. The terms that do not depend on temperature will vanish.