## Energy and Entropy: Fall-2020HW 3: Due Friday 10/16

1. Rubber Sheet

Consider a hanging rectangular rubber sheet. We will consider there to be two ways to get energy into or out of this sheet: you can either stretch it vertically or horizontally. The distance of vertical stretch we will call $y$, and the distance of horizontal stretch we will call $x$.

If I pull the bottom down by a small distance $\Delta y$, with no horizontal force, what is the resulting change in width $\Delta x$? Express your answer in terms of partial derivatives of the potential energy $U(x,y)$.

A diesel engine requires no spark plug. Rather, the air in the cylinder is compressed so highly that the fuel ignites spontaneously when sprayed into the cylinder.

In this problem, you may treat air as an ideal gas, which satisfies the equation $pV = Nk_BT$. You may also use the property of an ideal gas that the internal energy depends only on the temperature $T$, i.e. the internal energy does not change for an isothermal process. For air at the relevant range of temperatures the heat capacity at fixed volume is given by $C_V=\frac52Nk_B$, which means the internal energy is given by $U=\frac52Nk_BT$.

Note: in this problem you are expected to use only the equations given and fundamental physics laws. Looking up the formula in a textbook is not considered a solution at this level.

1. If the air is initially at room temperature (taken as $20^{o}C$) and is then compressed adiabatically to $\frac1{15}$ of the original volume, what final temperature is attained (before fuel injection)?

2. By what factor does the pressure increase?

3. Bottle in a Bottle

The internal energy of helium gas at temperature $T$ is to a very good approximation given by \begin{align} U &= \frac32 Nk_BT \end{align}

Consider a very irreversible process in which 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. What is the change in temperature when this process is complete? How much of the helium will remain in the small bottle?

4. Microwave ice calorimetry lab questions These questions go with the microwave version of the ice calorimetry lab. This is not a lab report, but I do expect you to explain your data and your analysis in each result
1. What is the specific latent heat of fusion of water in terms of microwave oven seconds or minutes. Use the power of your microwave oven to convert this into Joules per gram.
2. What is the specific latent heat of vaporization of water in terms of microwave oven seconds or minutes. Use the power of your microwave oven to convert this into Joules per gram.
3. How does the ratio of these two latent heats, and how does it compare with the accepted value? Is the ratio more accurate than the latent heats themselves?
4. Optional If you had access to data taken with a thermometer, what is the specific heat of water at some temperature that you were able to measure? How does this compare with experiment when taken as a ratio of one of your latent heats?
5. Solve for the difference in entropy between cold liquid water at $0^\circ$C and ice at the same temperature. You may use either the accepted experimental values or your experimental value (in funny microwave oven units).
6. Solve for the difference in entropy between hot liquid water at $100^\circ$C and steam at the same temperature. You may use either the accepted experimental values or your experimental value (in funny microwave oven units).
7. Solve for the entropy difference between the cold and hot water mentioned above. You may assume that the heat capacity of liquid water is independent of temperature (which is not in general true), and may again use either the accepted measurements or your own oven units.
8. Sketch to scale the three entropy differences you've measured. What is the order of the four entropies in question?