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

1. Bottle in a Bottle 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, which is insulated.

The volume of the small bottle is 0.001 m23 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 does the large bottle contain? What is the final temperature of the gas?

2. Compute the integral $\int \frac{{\mathit{\unicode{273}}} Q}{T}$ and the change of entropy $\Delta S$ between the initial state (gas in the small bottle) and the final state (gas leaked in the big bottle).

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 activity in class where we put ice into some warm water. We started with 131 g of ice and added 205 g of water at 40°C. We then insulated the ice and water with a cozy blanket on a cozy chair. In class you worked out how much ice would be left when the water had cooled to 0°C. You may need to reproduce that work to complete this problem. We treated the specific heat of water as 4.18 J/g/K independently of temperature (which is an approximation), and the latent heat of melting as 333 J/g.
1. Change in entropy of water Work out the change in entropy of the water that happened as it cooled down.
2. Change in entropy of ice Work out the change in entropy of the ice as it was melted.
3. Net change What is the net change of entropy for this entire process in which no heat was exchanged with the environment?
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.