Biological Physics/The Third Law

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So far, we have encountered the first two laws of thermodynamics, and now it is time to see the third. For an ideal gas, recall how entropy, energy, and temperature are related dS = \frac{dU}{T} where temperature is held constant. Now when volume is also held constant, work on the system W = P \Delta V = 0, so dS = \frac{dQ}{T}. By the definition of heat in terms of heat capacity dS = \frac{C dT}{T}. To add up all the small incremental dSs, \Delta S = \int_{T_{i}}^{T_{f}} \frac{CdT}{T}.

Now, what happens as T_{i} = 0? Then, \Delta S =  \int_{0}^{T_{f}} \frac{CdT}{T}, where if C is not dependent on temperature, \Delta S = C \int_{0}^{T_{f}} \frac{dT}{T}. Then, \Delta S = C*ln(T_{f}) - ln(0). Since ln(0) approaches -∞, then this means that \Delta S gets infinitely large as T \rightarrow 0, which is not good! So, there must be some temperature dependence in the heat capacity C.

The Third Law of Thermodynamics states that heat capacity C must go to zero faster than ln(0) goes to infinity, implying \Delta S \rightarrow 0. So, the multiplicity Ω where S = k_{B}ln(\Omega) must be ln(\Omega) = S(0K) = 0 which implies \Omega = e^{0} = 1. So a more intuitive way to state the third law would be

There's only one configuration for a system at 0 K.

Playing the Game[edit]

A fun way to remember the Three Laws of Thermodynamics is with the following way:

0th Law

- You must play the game

1st Law

- You can't win the game

2nd Law

- You can't even break even

3rd Law

- Not even on a cold day

And there you have it: the Three Laws of Thermodynamics!