Thermodynamics and Statistical Mechanics

1
Thermodynamics and Statistical Mechanics

• Open Systems
• and
• Chemical Potential

2
Diffusive Interaction

• If particles are added to a system, the energy of
  the system can change, because of the chemical
  potential of the added particles in their new
  environment. A term is needed to account for this
  effect.
• dU TdS PdV mdn

3
Chemical Potential

• dU TdS PdV mdn
• In this equation, m is the chemical energy per
  kilomole, and dn is the change in the number of
  kilomoles.

4
Chemical Potential

• Suppose 2.0 10-5 kilomoles of acid is added to a
  1.0 liter of water at room temperature. The
  temperature of the water rises 0.15ºC. From this
  data the chemical potential of the acid in water
  can be calculated.

5
Chemical Potential
6
Chemical Potential
7
Chemical Potential
8
More Than One Component
9
Gibbs Function
10
Equilibrium Conditions
Consider two systems, A1 and A2, that can
interact thermally, mechanically, and
diffusively. For either system,
11
Equilibrium Conditions
The change in entropy for the combined system is
given by, DS0 DS1 DS2, where DS1 and DS2 are
given by the expression on the previous slide.
Then,
12
Equilibrium Conditions
Since the two systems are interacting only with
each other, we have, DU2 DU1 DV2 DV1 Dn2
Dn1
13
Equilibrium Conditions
14
Equilibrium Conditions

• When the two systems come to equilibrium, S0 will
  be a maximum. That means that DS0 will be zero
  for any small variations of U1, V1, or n1. That
  is possible only if the coefficients of DU1, DV1,
  and Dn1 are all zero.

15
Equilibrium Conditions
16
Approach to Equilibrium

• To examine the approach to equilibrium, we shall
  replace DU1 by Q1. To do so, use
• DU1 Q1 P1DV1 m1Dn1. Then,

17
Approach to Equilibrium

• DS0 gt 0, so each term must be positive.

If T1 gt T2 , DQ1 lt 0 If P1 gt P2 , DV1 gt 0 If
m1 gt m2 , Dn1 lt 0
18
Approach to Equilibrium

• When two systems interact,
• Heat flows from the hotter to the cooler.
• The system at higher pressure expands at the
  expense of the other.
• Particles flow from the system of higher chemical
  potential to the other system.