Chapter 1: Energy in Thermal Physics

In Problem 1.16 you calculated the pressure of the earth’s atmosphere as a function of altitude, assuming constant temperature. Ordinarily, however, the temperature of the bottommost 10-15 km of the atmosphere (called the troposphere) decreases with increasing altitude, due to heating from the ground (which is warmed by sunlight). If the temperature gradient $\left|dT/dz\right|$exceeds a certain critical value, convection will occur: Warm, low-density air will rise, while cool, high-density air sinks. The decrease of pressure with altitude causes a rising air mass to expand adiabatically and thus to cool. The condition for convection to occur is that the rising air mass must remain warmer than the surrounding air despite this adiabatic cooling.

a. Show that when an ideal gas expands adiabatically, the temperature and pressure are related by the differential equation

$\frac{\mathbf{d}\mathbf{T}}{\mathbf{d}\mathbf{P}}\mathbf{=}\frac{\mathbf{2}}{\mathbf{f}\mathbf{+}\mathbf{2}}\frac{\mathbf{T}}{\mathbf{P}}$

b. Assume that $\mathit{d}\mathit{T}\mathbf{/}\mathit{d}\mathit{z}$is just at the critical value for convection to begin so that the vertical forces on a convecting air mass are always approximately in balance. Use the result of Problem 1.16(b) to find a formula for $\mathit{d}\mathit{T}\mathbf{/}\mathit{d}\mathit{z}$in this case. The result should be a constant, independent of temperature and pressure, which evaluates to approximately $\mathbf{–}\mathbf{10}\mathbf{°}\mathit{C}\mathbf{/}\mathit{k}\mathit{m}$. This fundamental meteorological quantity is known as the dry adiabatic lapse rate.

Problem 1.41. To measure the heat capacity of an object, all you usually have to do is put it in thermal contact with another object whose heat capacity you know. As an example, suppose that a chunk of metal is immersed in boiling water (100°C), then is quickly transferred into a Styrofoam cup containing 250 g of water at 20°C. After a minute or so, the temperature of the contents of the cup is 24°C. Assume that during this time no significant energy is transferred between the contents of the cup and the surroundings. The heat capacity of the cup itself is negligible.

1. How much heat is lost by the water?
2. How much heat is gained by the metal?
3. What is the heat capacity of this chunk of metal?
4. If the mass of the chunk of metal is 100 g, what is its specific heat capacity?

The specific heat capacity of Albertson's Rotini Tricolore is approximately 1.8 J/g ^oC . Suppose you toss 340 g of this pasta (at 25^oC ) into 1.5 liters of boiling water. What effect does this have on the temperature of the water (before there is time for the stove to provide more heat)?

Calculate the heat capacity of liquid water per molecule, in terms of K . Suppose (incorrectly) that all the thermal energy of water is stored in quadratic degrees of freedom. How many degrees of freedom would each molecule have to have?

At the back of this book is a table of thermodynamic data for selected substances at room temperature. Browse through the $C_P$values in this table, and check that you can account for most of them (approximately) using the equipartition theorem. Which values seem anomalous?

Measured heat capacities of solids and liquids are almost always at constant pressure, not constant volume. To see why, estimate the pressure needed to keep $V$fixed as $T$increases, as follows.

(a) First imagine slightly increasing the temperature of a material at constant pressure. Write the change in volume,$d{V}_1$, in terms of $dT$and the thermal expansion coefficient $\beta$introduced in Problem 1.7.

(b) Now imagine slightly compressing the material, holding its temperature fixed. Write the change in volume for this process, $d{V}_2$, in terms of $dP$and the isothermal compressibility ${\kappa }_T$, defined as

${\kappa }_T\equiv -\frac{1}{V}{\left(\frac{\mathrm{\partial }V}{\mathrm{\partial }P}\right)}_T$

(c) Finally, imagine that you compress the material just enough in part (b) to offset the expansion in part (a). Then the ratio of $dP\text{to}dT$is equal to $(\mathrm{\partial }P/\mathrm{\partial }T{)}_V$, since there is no net change in volume. Express this partial derivative in terms of $\beta \text{and}{\kappa }_T$. Then express it more abstractly in terms of the partial derivatives used to define $\beta \text{and}{\kappa }_T$. For the second expression you should obtain

${\left(\frac{\mathrm{\partial }P}{\mathrm{\partial }T}\right)}_V=-\frac{(\mathrm{\partial }V/\mathrm{\partial }T{)}_P}{(\mathrm{\partial }V/\mathrm{\partial }P{)}_T}$

This result is actually a purely mathematical relation, true for any three quantities that are related in such a way that any two determine the third.

(d) Compute $\beta ,{\kappa }_T,\text{and}(\mathrm{\partial }P/\mathrm{\partial }T{)}_V$for an ideal gas, and check that the three expressions satisfy the identity you found in part (c).

(e) For water at ${25}^{\circ }\mathrm{C},\beta =2.57\times {10}^{-4}{\mathrm{K}}^{-1}\text{and}{\kappa }_T=4.52\times {10}^{-10}{\mathrm{Pa}}^{-1}$. Suppose you increase the temperature of some water from ${20}^{\circ }\mathrm{C}\text{to}{30}^{\circ }\mathrm{C}$. How much pressure must you apply to prevent it from expanding? Repeat the calculation for mercury, for which $\text{(at}{25}^{\circ }\mathrm{C}\text{)}\beta =1.81\times {10}^{-4}{\mathrm{K}}^{-1}$and${\kappa }_T=4.04\times {10}^{-11}{\mathrm{Pa}}^{-1}$

Given the choice, would you rather measure the heat capacities of these substances at constant $v$or at constant $p$?

Your $200g$cup of tea is boiling-hot. About how much ice should you add to bring it down to a comfortable sipping temperature of ${65}^{\circ }\mathrm{C}$? (Assume that the ice is initially at $-{15}^{\circ }\mathrm{C}$. The specific heat capacity of ice is $(\left.0.5\mathrm{cal}/\mathrm{g}{\cdot }^{\circ }\mathrm{C}.\right)$.)

Your 200 g cup of tea is boiling-hot. About how much ice should you add to bring it down to a comfortable sipping temperature of $65°\mathrm{C}$? (Assume that the ice is initially $65°\mathrm{C}$. The specific heat capacity of ice isrole="math" localid="1650146844935" $0.5\mathrm{cal}/\mathrm{g}°\mathrm{C}$.

When spring finally arrives in the mountains, the snow pack may be two meters deep, composed $50\mathrm{\%}$of ice and $50\mathrm{\%}$of air. Direct sunlight provides about $1000{\text{watts/m}}^2$to earth's surface, but the snow might reflect $90\mathrm{\%}$of this energy. Estimate how many weeks the snow pack should last, if direct solar radiation is the only source of energy.

Problem 1.49. Consider the combustion of one mole of ${\mathrm{H}}_2$with$1/2$ mole of${\mathrm{O}}_2$ under standard conditions, as discussed in the text. How much of the heat energy produced comes from a decrease in the internal energy of the system, and how much comes from work done by the collapsing atmosphere? (Treat the volume of the liquid water as negligible.)