Energy and Entropy: Winter-2026 HW 3: Due Day 10 Tues 10/8
Name the Experiment
Consider the following derivative:
\begin{align}
\left(\frac{\partial {U}}{\partial {p}}\right)_{S,N}
\end{align}
Design an experiment to measure this derivative. In your answer, include a schematic of the aperatus and label the quantitites you would measure. Describe your measurement process, and show the algebra required to convert your directly measured quantities into the derivative.
Extensive Internal Energy
Consider a system which has an internal energy \(U\) defined by:
\begin{align}
U &= \gamma V^\alpha S^\beta
\end{align}
where \(\alpha\), \(\beta\) and \(\gamma\) are constants. The internal
energy is an extensive quantity. What constraint does this place on
the values \(\alpha\) and \(\beta\) may have?
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 vertical dimension of the rubber sheet we will call \(y\), and the horizontal dimension of the rubber sheet we will call \(x\). We can use these two independent variables to specify the "state" of the rubber sheet. Similiar to the partial derivative machine, we could choose any pair of variables from the set \(\{ x,y,F_x,F_y \}\) to specify the state of the rubber sheet.
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 \(U(x,y)\).
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? What fraction of the
helium will remain in the small bottle?
