For each of the following equations, check whether it could possibly make sense. You will need to check both dimensions and whether the quantities involved are intensive or extensive. For each equation, explain your reasoning.
You may assume that quantities with subscripts such as \(V_0\) have the same dimensions and intensiveness/extensiveness as they would have without the subscripts.
\[p = \frac{N^2k_BT}{V}\]
\[p = \frac{Nk_BT}{V}\]
\[U = \frac32 k_BT\]
\[U = - Nk_BT \ln\frac{V}{V_0}\]
\[S = - k_B \ln\frac{V}{V_0}\]
Consider a one-dimensional object such as a stretched rubber band. Categorize as intensive or extensive its properties of length, tension, mass and internal energy.
Translate the information on this diagram from the T-V plane to the p-V plane (i.e. draw contours of constant \(T\) on a graph of \(p\) vs \(V\)). Include point \(A\) on your p-V graph. Complete your graph by hand using discrete data points that you read from the T-V diagram. Make a fairly accurate sketch of the contours using the attached grid or in some other way making nice square axes with appropriate tick marks. Don't make up data for pressures above 1000 Pa or below 400 Pa.
Are the lines that you drew straight or curved? What feature of the \(TV\) graph would have to change to change this result?
Sketch the line of constant temperature that passes through the point A.
Find the differential of each of the following expressions; zap each of the following with \(d\). Note that all italicized letters are variables:
\[f=3x-5z^2+2xy\]
\[g=\frac{c^{1/2}b}{a^2}\]
\[h=\sin^2(\omega t)\]
\[j=a^x\]