Chapter 3: Interactions and Implications

What partial-derivative relation can you derive from the thermodynamic identity by considering a process that takes place at constant entropy? Does the resulting equation agree with what you already knew? Explain.

A liter of air, initially at room temperature and atmospheric pressure, is heated at constant pressure until it doubles in volume. Calculate the increase in its entropy during this process.

Sketch a qualitatively accurate graph of the entropy of a substance (perhaps${\mathrm{H}}_2\mathrm{O}$ ) as a function of temperature, at fixed pressure. Indicate where the substance is solid, liquid, and gas. Explain each feature of the graph briefly.

Figure 3.3 shows graphs of entropy vs. energy for two objects, A and B. Both graphs are on the same scale. The energies of these two objects initially have the values indicated; the objects are then brought into thermal contact with each other. Explain what happens subsequently and why, without using the word "temperature."

As shown in Figure 1.14, the heat capacity of diamond near room temperature is approximately linear in T. Extrapolate this function up to $500\mathrm{K}$, and estimate the change in entropy of a mole of diamond as its temperature is raised from$298\mathrm{K}$ to $500\mathrm{K}$. Add on the tabulated value at$298\mathrm{K}$ (from the back of this book) to obtain $S(500\mathrm{K})$.

Experimental measurements of heat capacities are often represented in reference works as empirical formulas. For graphite, a formula that works well over a fairly wide range of temperatures is (for one mole)

$C_P=a+bT-\frac{c}{T^2}$

where $a=16.86\mathrm{J}/\mathrm{K},b=4.77\times {10}^{-3}\mathrm{J}/{\mathrm{K}}^2$, and $c=8.54\times {10}^5\mathrm{J}·\mathrm{K}$. Suppose, then, that a mole of graphite is heated at constant pressure from $298\mathrm{K}$to $500\mathrm{K}$. Calculate the increase in its entropy during this process. Add on the tabulated value of $S(298\mathrm{K})$(from the back of this book) to obtain $S(500\mathrm{K})$.

A cylinder contains one liter of air at room temperature ( $300\mathrm{K}$) and atmospheric pressure $\left({10}^5\mathrm{N}/{\mathrm{m}}^2\right)$. At one end of the cylinder is a massless piston, whose surface area is $0.01{\mathrm{m}}^2$. Suppose that you push the piston in very suddenly, exerting a force of $2000\mathrm{N}$. The piston moves only one millimeter, before it is stopped by an immovable barrier of some sort.

(a) How much work have you done on this system?

(b) How much heat has been added to the gas?

(c) Assuming that all the energy added goes into the gas (not the piston or cylinder walls), by how much does the internal energy of the gas increase?

(d) Use the thermodynamic identity to calculate the change in the entropy of the gas (once it has again reached equilibrium).

Use the thermodynamic identity to derive the heat capacity formula

$C_V=T{\left(\frac{\partial S}{\partial T}\right)}_V$

which is occasionally more convenient than the more familiar expression in terms of $U$. Then derive a similar formula for $C_P$, by first writing $dH$in terms of $dS$and $dP$.

In the text I showed that for an Einstein solid with three oscillators and three units of energy, the chemical potential is $\mu =-ϵ$(where $ϵ$is the size of an energy unit and we treat each oscillator as a "particle"). Suppose instead that the solid has three oscillators and four units of energy. How does the chemical potential then compare to $-ϵ$ ? (Don't try to get an actual value for the chemical potential; just explain whether it is more or less than $-ϵ$.)

Consider an Einstein solid for which both N and q are much greater than $1$. Think of each oscillator as a separate "particle."

(a) Show that the chemical potential is

role="math" localid="1646995468663" $\mu =-kT\ln \left(\frac{N+q}{N}\right)$

(b) Discuss this result in the limits $N\gg q$and $N\ll q$, concentrating on the question of how much $S$increases when another particle carrying no energy is added to the system. Does the formula make intuitive sense?