Statistical thermodynamics Lectures 7, 8 Quantum Classical Energy levels Bulk properties Various forms of energies. Everything turns out to be controlled by temperature CY1001 T. Pradeep Ref. Atkins 9 th edition Alberty, Silbey, Bawendi 4 th edition
Need for statistical thermodynamics Microscopic and macroscopic world Energy states Distribution of energy - population Principle of equal a priori probabilities ε 0 Reference to zero Configuration - instantaneous n 1, n 2, molecules exist in states with energies ε 0, ε 1, N is the total number of molecules {N,0,0, } and {N-2, 2,0, } are configurations Second is more likely than the first Fluctuations occur {1,1}, {2,0}, {0,2} Weight of a configuration = how many times the configuration can be reached.
Energy Generalized picture of weight A configuration {N-2,2,0,0, } First ball of the higher state can be chosen in N ways, because there are N balls Second ball can be chosen in N-1 ways as there are N-1 balls But we need to avoid A,B from B,A. Thus total number of distinguishable configurations is, ½ [N(N-1)] 1 2 3 4 5 6 7 8 9 10 0 A configuration {3,2,0,0,..} is chosen in 10 different ways - weight of a configuration How about a general case of N particles? {n 0,n 1,..} configuration of N particles. Distinct ways, W = N! n 0!n 1!n 2!... N! ways of selecting balls (first ball N, second (N-1), etc.) n 0! ways of choosing balls in the first level. n 1! for the second, etc. W is the weight of the configuration. How many ways a configuration can be achieved.
Better to use natural logarithm ln W = ln N! n 0! n 1! n 2!... = ln N!- ln(n 0! n 1! n 2!..) =ln N! -(ln n 0! +ln n 1! +ln n 2! +...) =ln N! -Σ i ln n i! ln x! x ln x-x Stirling s approximation ln W = (N ln N - N) - Σ i (n i ln n i n i ) = N ln N Σ i n i ln n i We have used, Σ i n i = N
Which is the dominating configuration having maximum weight? Generalized approach is to set dw = 0 There are problems as any configuration is not valid. 1. Energy is constant. Σ i N i ε i = E 2. Constant number of molecules. Σ i N i = N W N i Populations in the configuration of greatest weight depends on the energy of the state. N i N = e -βε i Σ i e -βε i β = 1 kt Boltzmann distribution i is a sum over available states Temperature gives the most probable populations
Boltzmann distribution population, p i = e -βε i q Molecular partition function Another form of q: Look at limiting cases q =Σ i e -βε i q =Σ levels l g l e -βε i lim T 0 q = g 0 lim T = When number of states is finite, we will not get. There are several ways of looking at i How to look at partition functions? 1. Partition function is the number of available states. 2. Partition function is the number of thermally accessible states. 3. How molecules are partitioned into available states. How to look at thermodynamic properties? Because, ε 0 = 0 For all higher levels ε is finite. e -βε =1. Because, e -x is 0 when x is. All terms will reduce to 1. e -x is 1 when x is 0
Evaluation of molecular partition function 1 + x + x 2 + x 3 +.= 1/(1-x) q = 1 + e βε + e 2βε + e 3βε + = 1 + e βε + (e βε ) 2 + (e βε ) 2 + =1/(1- e βε ) 2ε ε 0 ε Fraction of molecules in energy levels is, p i = e βε i /q = (1- e βε ) e βε i Discussion of figure, next slide For a two level system, p p o = 1/(1 + e βε ) p 1 = e βε /q = e βε / (1+ e βε ) As T à, both p o and p 1 go to ½. Consequences?
Low temperature High temperature
Approximations: Partition functions are often complex. Analytical expressions cannot be obtained in several cases. Approximations become very useful. For example, partition function for one dimensional motion E n = n 2 h 2 ε n = (n 2-1)ε 8 mx 2 n =1,2. ε = h2 8mX 2 q x = Σ n=1 e -(n 2-1)βε Lowest energy level has an energy of h 2 8mX 2 Energy with respect to this level is, q x = 1 e -(n 2-1)βε dn energy levels are close, sum becomes an integral No big difference, if we take the lower limit to 0 and replace n 2-1 to n 2. q x = 1 1/2 1/2 βε 0 e -x 2 dx = 1 1/2 βε 2 = 2 1/2 m X h 2 β Substitute, x 2 = n 2 βε, meaning dn = dx/(βε) 1/2 q x = 2 m h 2 β 1/2 X Substitute for ε
Independent motion in three dimensions ε n1n2n3 = ε n1 (X) + ε n2 (Y) +ε n3 (Z) Energy is a sum of independent terms q =Σ all n e βε(x) n1 βε(y) n2 -βε(z) n3 = Σ n1 e- βε(x) n1 Σ n2 e -βε(y) n2 Σ n3 e -βε(z) n3 = q x q y q z 3/2 q = 2 m h 2 β XYZ q = V Ʌ 3 Ʌ = h β 1/2 2 m = h (2 mkt) 1/2 Ʌ has dimensions of length, thermal wavelength J =kg m -2 s -2 Question: How many more quantum states will be accessible for 18 O 2 compared to 16 O 2, if it were to be confined in a box of 1 cm 3?
How to get thermodynamics? All information about the thermodynamic properties of the system is contained in the partition function. Thermal wavefunction. Total energy We know, E = Σ i n i ε i E = N q Σ i ε i e-βεi ε i e βεi = - d dβ e-βεi Most probable configuration is dominating. We use Boltzmann distribution. E = - N q Σ i d dβ U = U(0) + E e βεi = - N q U = U(0)- N q q β v U = U(0)- N lnq β d dβ Σ N dq i e-βεi = q dβ 1. All E s are relative E is the value of U relative to T = 0. v 2. Derivative w.r.t. β is partial as there are other parameters (such as V) on which energy depends on. Partition function gives internal energy of the system.
How to get entropy? For a process, change in internal energy, U = U(0) + E U = U(0) + Σ i n i ε i Internal energy changes occur due to change in populations (N i + dn i ) or energy states (ε i + dε i ). Consider a general case: For constant volume changes, du = Σ i ε i dn i du = du(0) + Σ i N i dε i + Σ i ε i dn i du = dq rev = TdS ds = du/t = k β Σ i ε i dn i ds = k Σ i ( lnw/ N i ) dn i + kασ i dn i β ε i = ( lnw/ N i ) + α Number of molecules do not change. Second term is zero. ds = k Σ i ( lnw/ N i ) dn i = k (dlnw) S = klnw From the derivation of Boltzmann distribution α and β are constants This can be rewritten (after some manipulations) to S(T) = [U(T) - U(0)]/T + Nk ln q