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# Statistical Thermodynamics

Entropy is the least understood quantity in science, as are the laws that govern it. Avoid the pitfalls and understand entropy better than most working scientists.

Now, you make coffee in the same thermos at 200 deg F and leave it on the desk in your room. About how long (in hours) does it take for the coffee to cool to 140 deg F?

**Assumptions and Details**

- The outside temperature varies with time according to \(T(t) = \left(75+ 15 \cos \omega t\right)\ \text{deg F}\) where \(\omega = 2\pi / 24\ \text{hr}^{-1}\).
- Your room is 75 deg F.
- Your coffee follows Newton's law of cooling: \(\partial_t T_\text{cup} \propto \left(T_\text{env} - T_\text{cup}\right)\).

If there are \(N\) gas molecules in the box, find the expected number of gas molecules in the hot side when the box reaches steady state.

**Assumptions and Details**

- \(N=100,000\)
- \(T_\text{High}=700\text{ K}\)
- \(T_\text{Low}=200\text{ K}\)

Suppose an enormous amount of black soot enters the atmosphere (perhaps from volcanos or nuclear weapons) that can be heated by the Sun directly. When the soot surrounded Earth reaches equilibrium, it has the stable temperature \(T_{soot}\).

Find \(T_{soot}/T_0\), where \(T_0\) is the temperature of the Earth with no atmosphere whatsoever.

**Assumptions**

- The surface of the Earth and the soot layer are perfect blackbodies.
- Approximate the soot as a spherical shell around the Earth.

Concretely, each cycle drives some heat \(\delta Q\) from the hot reservoir to the cold reservoir, and the engine performs some work \(\delta W\) to propel the car, such that the engine returns to its original state after each cycle. Clearly, each cycle lowers the temperature \(T_+\), and raises the temperature \(T_-\), until \(T_+=T_-=T^*\), at which point the engine can do no further work.

Find \(T^*\) (in deg Kelvin) at the point where the engine stops running.

**Assumptions and Details**

- The entire process is thermodynamically reversible.
- \(T_+\) = 423\(^\circ\) K
- \(T_-\) = 300\(^\circ\) K
- Both heat reservoirs are metal blocks of heat capacity \(\gamma\)

Hint: The first **and** second laws of thermodynamics are your friend.

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