Thermo & Stat Mech - Spring 2006 Class 19 1 Thermodynamics and Statistical Mechanics Partition Function

Thermo & Stat Mech - Spring 2006 Class 192 Free Expansion of a Gas

Thermo & Stat Mech - Spring 2006 Class 193 Free Expansion

Thermo & Stat Mech - Spring 2006 Class 194 Isothermal Expansion

Thermo & Stat Mech - Spring 2006 Class 195 Isothermal Expansion Reversible route between same states. đQ = đW + dU Since T is constant, dU = 0. Then, đQ = đW.

Thermo & Stat Mech - Spring 2006 Class 196 Entropy Change The entropy of the gas increased. For the isothermal expansion, the entropy of the Reservoir decreased by the same amount. So for the system plus reservoir,  S = 0 For the free expansion, there was no reservoir.

Thermo & Stat Mech - Spring 2006 Class 197 Statistical Approach

Thermo & Stat Mech - Spring 2006 Class 198 Statistical Approach

Thermo & Stat Mech - Spring 2006 Class 199 Partition Function

Thermo & Stat Mech - Spring 2006 Class 1910 Boltzmann Distribution

Thermo & Stat Mech - Spring 2006 Class 1911 Maxwell-Boltzmann Distribution Correct classical limit of quantum statistics is Maxwell-Boltzmann distribution, not Boltzmann. What is the difference?

Thermo & Stat Mech - Spring 2006 Class 1912 Maxwell-Boltzmann Probability w B and w MB yield the same distribution.

Thermo & Stat Mech - Spring 2006 Class 1913 Relation to Thermodynamics

Thermo & Stat Mech - Spring 2006 Class 1914 Relation to Thermodynamics

Thermo & Stat Mech - Spring 2006 Class 1915 Chemical Potential dU = TdS – PdV +  dN In this equation,  is the chemical energy per molecule, and dN is the change in the number of molecules.

Thermo & Stat Mech - Spring 2006 Class 1916 Chemical Potential

Thermo & Stat Mech - Spring 2006 Class 1917 Entropy

Thermo & Stat Mech - Spring 2006 Class 1918 Entropy

Thermo & Stat Mech - Spring 2006 Class 1919 Helmholtz Function

Thermo & Stat Mech - Spring 2006 Class 1920 Chemical Potential

Thermo & Stat Mech - Spring 2006 Class 1921 Chemical Potential

Thermo & Stat Mech - Spring 2006 Class 1922 Boltzmann Distribution

Thermo & Stat Mech - Spring 2006 Class 1923 Distributions

Thermo & Stat Mech - Spring 2006 Class 1924 Distributions

Thermo & Stat Mech - Spring 2006 Class 1925 Ideal Gas

Thermo & Stat Mech - Spring 2006 Class 1926 Ideal Gas

Thermo & Stat Mech - Spring 2006 Class 1927 Ideal Gas

Thermo & Stat Mech - Spring 2006 Class 1928 Entropy

Thermo & Stat Mech - Spring 2006 Class 1929 Math Tricks For a system with levels that have a constant spacing (e.g. harmonic oscillator) the partition function can be evaluated easily. In that case,  n = n , so,

Thermo & Stat Mech - Spring 2006 Class 1930 Heat Capacity of Solids Each atom has 6 degrees of freedom, so based on equipartition, each atom should have an average energy of 3kT. The energy per mole would be 3RT. The heat capacity at constant volume would be the derivative of this with respect to T, or 3R. That works at high enough temperatures, but approaches zero at low temperature.

Thermo & Stat Mech - Spring 2006 Class 1931 Heat Capacity Einstein found a solution by treating the solid as a collection of harmonic oscillators all of the same frequency. The number of oscillators was equal to three times the number of atoms, and the frequency was chosen to fit experimental data for each solid. Your class assignment is to treat the problem as Einstein did.

Thermo & Stat Mech - Spring 2006 Class 1932 Heat Capacity