Bottle in a Bottle Part 2

  • heat entropy ideal gas
  • Energy and Entropy 2020

    Consider the bottle-in-a-bottle problem in a previous problem set, summarized here.

    A small bottle of helium is placed inside a large bottle, which otherwise contains vacuum. The inner bottle contains a slow leak, so that the helium leaks into the outer bottle. The inner bottle contains one tenth the volume of the outer bottle. The outer bottle is insulated.

    The volume of the small bottle is 0.001 m3 and the volume of the big bottle is 0.01 m3. The initial state of the gas in the small bottle was \(p=106\) Pa and its temperature \(T=300\) K. Approximate the helium gas as an ideal gas of equations of state \(pV=Nk_BT\) and \(U=\frac32 Nk_BT\).

    1. How many molecules of gas are initially in the small bottle? What is the final temperature of the gas after the pressures have equalized?

    2. Compute the change of entropy \(\Delta S\) between the initial state (gas in the small bottle) and the final state (gas in both bottles, pressures equalized). Do not use the Sackur-Tetrode equation, use an alternative method.

    3. Discuss your results.