This is a review of Thermodynamics and Statistical Mechanics.
Thermodynamic Laws/Definition of Entropy • 1st law of thermodynamics, the conservation of energy:
dU = dq − dw ∆ = Q − W, (1)
where dq is heat entering the system and dw is work done by the system. Note the convention: (+) for energy entering the system and (-) for energy leaving the system. • 2nd law of thermodynamics, entropy: In any spontaneous transition, the entropy of the universe increases.
There are many equivalent statements of the second law, such as: heat never flows from cold to hot, or, there is no such thing as a perpetual motion machine. In a reversible transformation, the entropy of the universe does not change. Note: this does not mean that the entropy of some sub-universal system will not increase or decrease. It is important to consider all parts of your system. To find the change in entropy of a system between state A and state B, connect A and B by a reversible path. Then Z B dq S(B) − S(A) = . (2) A T Using Equation 2, we can illustrate the point about reversible transformations above: Consider a reversible isothermal expansion of an ideal gas in contact with a thermal reservoir 3 at temperature T . For an ideal gas, U = 2 kT , for an isothermal expansion dU = 0, and dq = P dV , so Q Z dq Z P dV = = ∆S = . (3) T T T nRT P = V , so Z P Z VB nR V dV = dV = nR ln B > 0, (4) T VA V VA −Q but heat leaves the reservoir, thus ∆Sres = T and ∆Suniverse = 0.
Carnot Cycle The Carnot Cycle is usually discussed with an ideal gas as the working substance, but in reality any thermodynamic system (a paramagnet, an electrochemical cell, etc.) can be used. A Carnot Cycle is a cycle involving two reversible isothermal transitions and two reversible adiabatic transitions. If the working substance is an ideal gas, the P − V diagram of the cycle looks like Fig. 1, and the S − T diagram looks like Fig. 2, The efficiency of the Carnot Cycle is given by W Q − Q T − T T η = = 2 1 = 2 1 = 1 − 1 , (5) Q2 Q2 T2 T2
1 Figure 1: Carnot Cycle: P − V diagram
Figure 2: Carnot Cycle: S − T diagram
which is work done by the system divided by the heat absorbed by the system. For a “Carnot Refrigerator,” run the cycle backwards:
Q1 Q1 1 1 T1 ηref = = = = = , (6) Q2 T2 W Q2 − Q1 − 1 − 1 T2 − T1 Q1 T1
heat extracted from the system divided by work required to do so. Note that as T1 → T2, ηref → ∞. In general, the efficiency is what you get out η = . (7) what you need to put in
Typical Phase Diagrams Figs. 3 and 4 show the phase diagrams for a single component system.
2 Figure 3: P − v diagram: Pressure vs. specific volume
Thermodynamic Potentials Let’s begin with Eq. 1: dU = dq − dw. (8) If we connect states by reversible processes we get T dS = dq and dW = P dV for a gas system. So, dU = T dS − P dV, (9) and setting one of ∂s or ∂v equal to zero yields,
∂U ∂U = T, = −P. (10) ∂S V ∂V S Define