Energy and Entropy: Fall-2020
HW 4: Due Friday 10/30

  1. Bottle in a Bottle Part 2

    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.

  2. Free Expansion

    The internal energy is of any ideal gas can be written as \begin{align} U &= U(T,N) \end{align} meaning that the internal energy depends only on the number of particles and the temperature, but not the volume.*

    The ideal gas law \begin{align} pV &= Nk_BT \end{align} defines the relationship between \(p\), \(V\) and \(T\). You may take the number of molecules \(N\) to be constant. Consider the free adiabatic expansion of an ideal gas to twice its volume. “Free expansion” means that no work is done, but also that the process is also neither quasistatic nor reversible.

    1. What is the change in entropy of the gas? How do you know this?

    2. What is the change in temperature of the gas?

  3. Melting ice lab questions These questions relate to the in-class activity where you put ice (prepared at zero degree celcius) into some warm water. You insulated the ice and water inside nested polystyrene cups and closed the lid. You waited a few minutes for the system to reach equilibrium. At equilibruium, all the melted ice and water were at the same temperature. During class, you started to calculate a prediction for this final temperature.
    1. Prediction What is your prediction for the final temperature? List the measured quantities that you used in your calcultion. Show your work.
    2. Measurement What final temperature did you measure? Comment on the magnitude and sources of errors in your experiment and in your prediction.
    3. Change in entropy Calculate the total change in entropy that occured during the overall process. Initial state: warm water and ice. Final state: cool water at a uniform temperature.
  4. PDM Elevator (Student Data) Collect data for an “elevator” cycle on your PDM and answer the following questions. (Note: You may find it useful to do calculations using a spreadsheet or another computer program. If you do so, please include both a printout of your calculations and any formulas that you used in the spreadsheet.)
    1. Make a table of data in a spreadsheet for your PDM that shows the value of each state variable at different points in an elevator cycle on the Partial Derivative Machine. Include at least 8 points for each separate process on the cycle, including the endpoints. You do NOT need to repeat the same data points multiple times.
    2. Numerically calculate the work done by each force (\(F_L\) and \(F_R\)) during each of the four processes (loading, raising, unloading, and lowering). Discuss your method for calculating the work: do not assume a functional form for the data. Include relevant graphs of the data and give a physical explanation for the sign of each work.
    3. What is the total energy of the system at the end of each process (loading, raising, unloading, and lowering)? I am asking for a number with units! Give a physical explanation for the total energy of the system after the elevator has returned to the initial state (ground floor, empty).
    4. The data you have used is from a real experiment you have done using the partial derivative machine --- and some of the data no doubt wasn't perfect! Discuss how you chose to handle this during your calculations.