Definition: Entropy
Entropy $S$ is a state variable that measures disorder. The increase in entropy over a reversible process is
$$dS = \frac{dQ}{T}.$$
However, because it is a state variable, even for non-reversible processes, you can find a reversible path on which you can use the formula.
Using this definition of entropy, we can also rewrite the first law of thermodynamics in the following form:
$$dU = TdS - pdV.$$
From this, you can define entropy as a function of $T$ and $V$, as you'll do in the problem set.
Definition: Heat Engines
Heat engines are systems that convert heat energy into work. They usually work between two temperature reservoirs,
Theorem: 2nd Law of Thermodynamics
The 2nd Law of Thermodynamics states that the entropy in the universe or any isolated system increases over time. A direct corollary of this is that heat flows from hotter systems to colder systems.
Theorem: Carnot Efficiency
The maximum achievable efficiency is that of a Carnot engine,
$$\eta = 1 - \frac{T_C}{T_H},$$
which can be proven by the second law of thermodynamics.
Idea: Kinetic Theory
Considering individual collisions between particles can derive thermodynamic results from first principle. In Kinetic theory, we make the following assumptions:
- The gas is composed of many identical particles of negligible size and moving randomly
- All collisions are elastic
Then, we consider collisions between particles or collisions between particles and walls to derive pressure, viscosity, and many other properties about gas.
Tip: Estimating
Often times in olympiad kinetic theory problems, you won't have to actually calculate things exactly -- you'll just have to get the expression up to an order of magnitude. So, if it's confusing, you can try dropping the constant dimensionless factors at the front of your expressions.
Theorem: Maxwell-Boltzmann Distribution
The probability distribution for the speeds of a given gas is
$$f(v) dv \propto v^2 e^{-mv^2/2k_BT}dv.$$
Integrating this distribution gives us the average and rms speeds:
$$\langle v \rangle = \sqrt{\frac{8k_BT}{m}}, \langle v^2 \rangle = \frac{3k_BT}{m}.$$
Idea: Effusion
Effusion is the process of particles leaking out of a small hole in a container. The whole should have characteristic length much smaller than that of the mean free path. Using kinetic theory, one can derive that the rate of effusion per unit area is
$$\Phi = \frac 14 n \langle v \rangle = \frac{p}{\sqrt{2\pi m k_B T}}.$$
Idea: Mean Free Path \& Time
As gas particles bounce around, they collide with one another. Using Kinetic theory, one can derive the characteristic time and sitance separating collision events:
$$\tau = \frac{1}{\sqrt 2 n\sigma \langle v\rangle},\lambda = \frac{1}{\sqrt 2 n \sigma}.$$