Let's choose a simple system: a gas in a piston. To hold the temperature fixed we want to put the piston in a thermal bath at temperature \(T\). To change the entropy, we need to add heat to the system. At first, this may seem counterintuitive: how do we heat something up while keeping the temperature constant? Well, heat and temperature are not the same thing, and adding heat will increase the total energy, not necessarily the temperature, and as long as we have our piston in a thermal bath and we add heat slowly the temperature will not be able to go up (what else will need to be happening?). So our experiment will involve adding a bit of heat to the gas, maybe by exposing one end of the piston to fire or embedding a resistor in the gas like we did in the lab. Since we need to know \(\Delta S\), we can relate it to the total heat \(Q\) using \(\Delta S = \frac{Q}{T}\) (the temperature is constant, so this equation is valid!). Then we measure the resulting change in volume, \(\Delta V\), by measuring the change in height of the piston, \(\Delta z\), and the area of the plunger that contacts the gas, \(A\). Putting it all together gives: \begin{align} \left(\frac{\partial {V}}{\partial {S}}\right)_{T} &\approx \frac{\Delta V}{\Delta S} = \frac{A\Delta z}{Q/T} \\ &= AT\frac{\Delta z}{Q} \end{align}
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.
\[p = \frac{N^2k_BT}{V}\]
This doesn't make sense, because \(N\) and \(V\) are extensive, while everything else is intensive, so the right hand side is extensive, but the left hand side is intensive.
\[p = \frac{Nk_BT}{V}\]
This does make sense, which isn't a surprise because it is the ideal gas law. Both sides are intensive, because the extensiveness of \(N\) and \(V\) cancels. In addition, the units work out right, because \(k_BT\) is energy, and pressure has units of energy over volume (which is the same as force over area).
\[U = \frac32 k_BT\]
This one doesn't make sense, because the internal energy \(U\) is extensive, but \(k_BT\) is intensive. They can't be equal.
\[U = - Nk_BT \ln\frac{V}{V_0}\]
This makes sense. \(Nk_BT\) is extensive and has dimensions of energy. \(\ln\frac{V}{V_0}\) is intensive and dimensionless, so all is good.
\[S = - k_B \ln\frac{V}{V_0}\]
This doesn't makes sense. \(S\) has the same dimensions as \(k_B\), and the log is all right, but the right hand side is intensive, while entropy is extensive.
\[S = - k_B \ln\frac{V}{N}\]
This is even worse. The logarithm of volume over number is a logarithm of a dimensionful quantity, which is very bad. The right hand side also doesn't match entropy in being extensive.