Energy in Thermal Physics

An ideal gas is made to undergo the cyclic process shown in the given figure. For each of the steps A, B, and C, determine whether each of the following is positive, negative, or zero: (a) the work done on the gas; (b) the change in the energy content of the gas; (c) the heat added to the gas. 

Then determine the sign of each of these three quantities for the whole cycle. What does this process accomplish?

An ideal diatomic gas, in a cylinder with a movable piston, undergoes the rectangular cyclic process shown in the given figure.

Assume that the temperature is always such that rotational degrees of freedom are active, but vibrational modes are "frozen out." Also assume that the only type of work done on the gas is quasistatic compression-expansion work.

(a) For each of the four steps A through D, compute the work done on the gas, the heat added to the gas, and the change in the energy content of the gas. Express all answers in terms of $P_1,P_2,V_1, and V_2$. (Hint: Compute $\Delta U$ before Q, using the ideal gas law and the equipartition theorem.)

(b) Describe in words what is physically being done during each of the four steps; for example, during step A, heat is added to the gas (from an external flame or something) while the piston is held fixed.

(c) Compute the net work done on the gas, the net heat added to the gas, and the net change in the energy of the gas during the entire cycle. Are the results as you expected? Explain briefly.

The Rankine temperature scale(abbreviated $°R$) uses the same scale size degrees as Fahrenheit, but measured up from absolute zero like Kelvin(so Rankine is to Fahrenheit as Kelvin is to Celsius). Find the conversion formula between Rankine and Fahrenheit and also between Rankine and Kelvin. What is the room temperature on the Rankine scale?

Does it ever make sense to say that one object is "twice as hot" as another? Does it matter whether one is referring to Celsius or Kelvin temperatures? Explain.

Determine the Kelvin temperature for each of the following:

(a) human body temperature;

(b) the boiling point of water(at the standard pressure of 1 atm);

(c) the coldest day u can remember;

(d) the boiling point of liquid nitrogen(-196$°$C);

(e) the melting point of lead(327$°$C) 

When you're sick with a fever and you take your temperature with a thermometer, approximately what is the relaxation time?

The Fahrenheit temperature scale is defined so that ice melts at 32⁰ F and water boils at 212⁰ F.

(a) Derive the formula for converting from Fahrenheit to Celsius and back 

(b) What is absolute zero on the Fahrenheit  scale?

Given an example to illustrate why you cannot accurately judge the temperature of an object by how hot or cold it feels to the touch?

When the temperature of liquid mercury increases by one degree Celsius (or one kelvin), its volume increases by one part in 550,000 . The fractional increase in volume per unit change in temperature (when the pressure is held fixed) is called the thermal expansion coefficient, β  :
$\beta \equiv \frac{\Delta V/V}{\Delta T}$
(where V is volume, T is temperature, and Δ signifies a change, which in this case should really be infinitesimal if β  is to be well defined). So for mercury, β =1 / 550,000 K^-1=1.81 x 10^-4 K^-1. (The exact value varies with temperature, but between 0^oC and 200^oC the variation is less than 1 %.)
(a) Get a mercury thermometer, estimate the size of the bulb at the bottom, and then estimate what the inside diameter of the tube has to be in order for the thermometer to work as required. Assume that the thermal expansion of the glass is negligible.
(b) The thermal expansion coefficient of water varies significantly with temperature: It is 7.5 x 10 ^-4 K^-1 at 100^oC, but decreases as the temperature is lowered until it becomes zero at 4^oC. Below 4^oC it is slightly negative, reaching a value of -0.68 x 10^-4K^-1 at 0^oC. (This behavior is related to the fact that ice is less dense than water.) With this behavior in mind, imagine the process of a lake freezing over, and discuss in some detail how this process would be different if the thermal expansion coefficient of water were always positive.

For a solid, we also define the linear thermal expansion coefficient, α, as the fractional increase in length per degree:

$\alpha \equiv \frac{\Delta L/L}{\Delta T}$
(a) For steel, α is 1.1 x 10^-5 K^-1. Estimate the total variation in length of a 1 km steel bridge between a cold winter night and a hot summer day.
(b) The dial thermometer in Figure 1.2 uses a coiled metal strip made of two different metals laminated together. Explain how this works.
(c) Prove that the volume thermal expansion coefficient of a solid is equal to the sum of its linear expansion coefficients in the three directions β=α_x + α_y + α_z. (So for an isotropic solid, which expands the same in all directions, β =3 α .)

A mole is approximately the number of protons in a gram of protons. The mass of a neutron is about the same as the mass of a proton, while the mass of an electron is usually negligible in comparison, so if you know the total number of protons and neutrons in a molecule(i,e., its "atomic mass"), you know the approximate mass(in grams) of a mole of these molecules. Referring to the periodic table at the back of this book ,find the mass of a mole of each of the following : Water, nitrogen (N₂), lead, quartz (S_i O₂)

Estimate the average temperature of the air inside a hot-air balloon (see Figure 1.1). Assume that the total mass of the unfilled balloon and payload is 500 kg. What is the mass of the air inside the balloon?

Even at low density, real gases don’t quite obey the ideal gas law. A systematic way to account for deviations from ideal behavior is the virial

expansion,

$\mathit{P}\mathit{V}\mathbf{-}\mathit{n}\mathit{R}\mathit{T}\mathbf{(}\mathbf{1}\mathbf{+}\frac{\mathbf{B}\left(T\right)}{\left(V/n\right)}\mathbf{+}\frac{\mathbf{C}\left(T\right)}{{\left(V/n\right)}^{\mathbf{2}}}\mathbf{+}\mathbf{\cdots }\mathbf{)}$

where the functions $\mathit{B}\mathbf{\left(}\mathbf{T}\mathbf{\right)}$, $\mathit{C}\mathbf{\left(}\mathbf{T}\mathbf{\right)}$, and so on are called the virial coefficients. When the density of the gas is fairly low, so that the volume per mole is large, each term in the series is much smaller than the one before. In many situations, it’s sufficient to omit the third term and concentrate on the second, whose coefficient $\mathit{B}\mathbf{\left(}\mathbf{T}\mathbf{\right)}$ is called the second virial coefficient (the first coefficient is 1). Here are some measured values of the second virial coefficient for nitrogen (${\mathit{N}}_{\mathbf{2}}$):

1. For each temperature in the table, compute the second term in the virial equation, $\mathit{B}\mathbf{\left(}\mathbf{T}\mathbf{\right)}\mathbf{/}\mathbf{(}\mathbf{V}\mathbf{/}\mathbf{n}\mathbf{)}$, for nitrogen at atmospheric pressure. Discuss the validity of the ideal gas law under these conditions.
2. Think about the forces between molecules, and explain why we might expect $\mathit{B}\mathbf{\left(}\mathbf{T}\mathbf{\right)}$ to be negative at low temperatures but positive at high temperatures.
3. Any proposed relation between $\mathit{P}$, $\mathit{V}$, and $\mathit{T}$, like the ideal gas law or the virial equation, is called an equation of state. Another famous equation of state, which is qualitatively accurate even for dense fluids, is the van der Waals equation,
   $\mathbf{(}\mathbf{P}\mathbf{+}\frac{\mathbf{a}{\mathbf{n}}^{\mathbf{2}}}{{\mathbf{V}}^{\mathbf{2}}}\mathbf{)}\mathbf{(}\mathbf{V}\mathbf{-}\mathbf{n}\mathbf{b}\mathbf{)}\mathbf{=}\mathit{n}\mathit{R}\mathit{T}$
   where a and b are constants that depend on the type of gas. Calculate the second and third virial coefficients ($\mathit{B}$ and $\mathit{C}$) for a gas obeying the van der Waals equation, in terms of $\mathit{a}$ and $\mathit{b}$. (Hint: The binomial expansion says that ${\mathbf{(}\mathbf{1}\mathbf{+}\mathbf{x}\mathbf{)}}^{\mathbf{p}}\mathbf{\approx }\mathbf{1}\mathbf{+}\mathit{p}\mathit{x}\mathbf{+}\frac{\mathbf{1}}{\mathbf{2}}\mathit{p}\mathbf{(}\mathbf{p}\mathbf{-}\mathbf{1}\mathbf{)}{\mathit{x}}^{\mathbf{2}}$, provided that $\mathbf{\left|}\mathbf{p}\mathbf{x}\mathbf{\right|}\mathbf{\ll }\mathbf{1}$. Apply this approximation to the quantity ${\mathbf{[}\mathbf{1}\mathbf{-}\left(nb/V\right)\mathbf{]}}^{\mathbf{-}\mathbf{1}}$.)
4. Plot a graph of the van der Waals prediction for $\mathit{B}\mathbf{\left(}\mathbf{T}\mathbf{\right)}$, choosing $\mathit{a}$ and $\mathit{b}$ so as to approximately match the data given above for nitrogen. Discuss the accuracy of the van der Waals equation over this range of conditions. (The van der Waals equation is discussed much further in Section 5.3.)

Calculate the rms speed of a nitrogen molecule at a room temperature?

Suppose you have a gas containing hydrogen molecules and oxygen molecules, in thermal equilibrium. Which molecule are moving faster, on average? By what factor?

Uranium has two common isotopes, with atomic masses of 238 and 235. one way to separate  these isotopes is to combine the uranium with fluorine to make uranium hexafluoride gas, UF_6,  then exploit the difference in the average thermal speeds of molecules containing the different isotopes. Calculate the rms speed of each molecule at room temperature, and compare them.

Calculate the rms speed of a nitrogen molecule at room temperature.

Suppose you have a gas containing hydrogen molecules and oxygen molecules, in thermal equilibrium. Which molecules are moving faster, on average? By what factor?

Uranium has two common isotopes, with atomic masses of 238 and 235. One way to separate these isotopes is to combine the uranium with fluorine to make uranium hexafluoride gas, UF₆, then exploit the difference in the average thermal speeds of molecules containing the different isotopes. Calculate the rms speed of each type of molecule at room temperature, and compare them.

During a hailstorm, hailstones with an average mass of 2g and a speed of 15 m/s strike a window pane at a 45^o angle. The area of the window is 0.5 m² and the hailstones hit it at a rate of 30 per second. What average pressure do they exert on the window? How does this compare to the pressure of the atmosphere?

If you poke a hole in a container full of gas, the gas will start leaking out. In this problem, you will make a rough estimate of the rate at which gas escapes through a hole. (This process is called effusion, at least when the hole is sufficiently small.)

1. Consider a small portion (area = $\mathit{A}$) of the inside wall of a container full of gas. Show that the number of molecules colliding with this surface in a time interval $\mathbf{\Delta }\mathit{t}$ is $\mathit{P}\mathit{A}\mathbf{\Delta }\mathit{t}\mathbf{/}\mathbf{\left(}\mathbf{2}\mathbf{m}\overline{{\mathbf{v}}_{\mathbf{x}}}\mathbf{\right)}$, where width="12" height="19" style="max-width: none; vertical-align: -6px;" $\mathit{P}$ is the pressure,  is the average molecular mass, and $\overline{{\mathbf{v}}_{\mathbf{x}}}$ is the average $\mathit{x}$ velocity of those molecules that collide with the wall.
   
2. It's not easy to calculate $\overline{{\mathbf{v}}_{\mathbf{x}}}$, but a good enough approximation is ${\left(\overline{v_x^2}\right)}^{\mathbf{1}\mathbf{/}\mathbf{2}}$, where the bar now represents an average overall molecule in the gas. Show that ${\left(\overline{v_x^2}\right)}^{\mathbf{1}\mathbf{/}\mathbf{2}}\mathbf{=}\sqrt{\mathbf{k}\mathbf{T}\mathbf{/}\mathbf{m}}$.
   
3. If we now take away this small part of the wall of the container, the molecules that would have collided with it will instead escape through the hole. Assuming that nothing enters through the hole, show that the number $\mathit{N}$ of molecules inside the container as a function of time is governed by the differential equation 
   $\frac{\mathbf{d}\mathbf{N}}{\mathbf{d}\mathbf{t}}\mathbf{=}\mathbf{-}\frac{\mathbf{A}}{\mathbf{2}\mathbf{V}}\sqrt{\frac{\mathbf{k}\mathbf{T}}{\mathbf{m}}\mathbf{N}}$
   Solve this equation (assuming constant temperature) to obtain a formula of the form $\mathit{N}\left(t\right)\mathbf{=}\mathit{N}\left(0\right){\mathit{e}}^{\mathbf{-}\mathbf{t}\mathbf{/}\mathbf{r}}$, where $\mathit{r}$ is the “characteristic time” for $\mathit{N}$ (and $\mathit{P}$) to drop by a factor of $\mathit{e}$.
   
4. Calculate the characteristic time for gas to escape from a 1-liter container punctured by a $\mathbf{1}\mathbf{-}{\mathbf{mm}}^{\mathbf{2}}$? hole.
5. Your bicycle tire has a slow leak so that it goes flat within about an hour after being inflated. Roughly how big is the hole? (Use any reasonable estimate for the volume of the tire.)
6. In Jules Verne’s Around the Moon, the space travelers dispose of a dog's corpse by quickly opening a window, tossing it out, and closing the window. Do you think they can do this quickly enough to prevent a significant amount of air from escaping? Justify your answer with some rough estimates and calculations.

Calculate the total thermal energy in a liter of helium at room temperature and atmospheric pressure. Then repeat the calculation for a liter of air. 

Calculate the total thermal energy in a gram of lead at room temperature, assuming that none of the degrees of freedom are "frozen out" (this happens to be a good assumption in this case). 

Calculate the total thermal energy in a liter of helium at room temperature and atmospheric pressure. Then repeat the calculation for a liter of air.

Calculate the total thermal energy in a gram of lead at room temperature, assuming that none of the degrees of freedom are "frozen out" (this happens to be a good assumption in this case).

List all the degrees of freedom, or as many as you can, for a molecule of water vapor. (Think carefully about the various ways in which the molecule can vibrate.)

A battery is connected in series to a resistor, which is immersed in water (to prepare a nice hot cup of tea). Would you classify the flow of energy from the battery to the resistor as "heat" or "work"? What about the flow of energy from the resistor to the water? 

A battery is connected in series to a resistor, which is immersed in water (to prepare a nice hot cup of tea). Would you classify the flow of energy from the battery to the resistor as "heat" or "work"? What about the flow of energy from the resistor to the water?

Give an example of a process in which no heat is added to a system, but its temperature increases. Then give an example of the opposite: a process in which heat is added to a system but its temperature does not change.

Estimate how long it should take to bring a cup of water to boiling temperature in a typical $600 -\mathrm{watt}$microwave oven, assuming that all the energy ends up in the water. (Assume any reasonable initial temperature for the water.) Explain why no heat is involved in this process. 

Put a few spoonfuls of water into a bottle with a tight lid. Make sure everything is at room temperature, measuring the temperature of the water with a thermometer to make sure. Now close the bottle and shake it as hard as you can for several minutes. When you're exhausted and ready to drop, shake it for several minutes more. Then measure the temperature again. Make a rough calculation of the expected temperature change, and compare. 

 Estimate how long it should take to bring a cup of water to boiling temperature in a typical 600 -watt microwave oven, assuming that all the energy ends up in the water. (Assume any reasonable initial temperature for the water.) Explain why no heat is involved in this process.

A cup containing 200g of water is sitting on your dining room table. After carefully measuring its temperature to be 20^oC, you leave the room. Returning ten minutes later, you measure its temperature again and find that it is now 25^oC. What can you conclude about the amount of heat added to the water? (Hint: This is a trick question.)

Put a few spoonfuls of water into a bottle with a tight lid. Make sure everything is at room temperature, measuring the temperature of the water with a thermometer to make sure. Now close the bottle and shake it as hard as you can for several minutes. When you're exhausted and ready to drop, shake it for several minutes more. Then measure the temperature again. Make a rough calculation of the expected temperature change, and compare.

Imagine some helium in a cylinder with an initial volume of $1litre$ and an initial pressure of $1 atm.$ Somehow the helium is made to expand to a final volume of $3 litres$, in such a way that its pressure rises in direct proportion to its volume.

(a) Sketch a graph of pressure vs. volume for this process.

(b) Calculate the work done on the gas during this process, assuming that there are no "other" types of work being done.

(c) Calculate the change in the helium's energy content during this process.

(d) Calculate the amount of heat added to or removed from the helium during this process.

(e) Describe what you might do to cause the pressure to rise as the helium expands.

Imagine some helium in a cylinder with an initial volume of 1 liter and an initial pressure of 1 atm. Somehow the helium is made to expand to a final volume of 3 liters, in such a way that its pressure rises in direct proportion to its volume.

1. Sketch a graph of pressure vs. volume for this process.
2. Calculate the work done on the gas during this process, assuming that there are no “other” types of work being done.
3. Calculate the change in the helium’s energy content during this process.
4. Calculate the amount of heat added to or removed from the helium during this process.
5. Describe what you might do to cause the pressure to rise as the helium expands.

By applying a pressure of $200 atm$, you can compress water to $99\%$ of its usual volume. Sketch this process (not necessarily to scale) on a $PV$ diagram, and estimate the work required to compress a liter of water by this amount. Does the result surprise you?

An ideal diatomic gas, in a cylinder with a movable piston, undergoes the rectangular cyclic process shown in Figure 1.10(b). Assume that the temperature is always such that rotational degrees of freedom are active, but vibrational modes are “frozen out.” Also, assume that the only type of work done on the gas is quasistatic compression-expansion work.

1.  For each of the four steps $\mathit{A}$ through $\mathit{D}$, compute the work done on the gas, the heat added to the gas, and the change in the energy content of the gas. Express all answers in terms of ${\mathit{P}}_{\mathbf{1}}$, ${\mathit{P}}_{\mathbf{2}}$, ${\mathit{V}}_{\mathbf{1}}$, and ${\mathit{V}}_{\mathbf{2}}$. (Hint: Compute $\mathbf{\Delta }\mathit{U}$ before $\mathit{Q}$, using the ideal gas law and the equipartition theorem.)
   
2. Describe in words what is physically being done during each of the four steps; for example, during step A, heat is added to the gas (from an external flame or something) while the piston is held fixed.
   
3. Compute the net work done on the gas, the net heat added to the gas, and the net change in the energy of the gas during the entire cycle. Are the results as you expected? Explain briefly.

Derive the equation 1.40 from the equation 1.39

In a Diesel engine, atmospheric air is quickly compressed to about 1/20 of its original volume. Estimate the temperature of the air after compression, and explain why a Diesel engine does not require spark plugs.

Two identical bubbles of gas form at the bottom of a lake, then rise to the surface. Because the pressure is much lower at the surface than at the bottom, both bubbles expand as they rise. However, bubble A rises very quickly, so that no heat is exchanged between it and the water. Meanwhile, bubble B rises slowly (impeded by a tangle of seaweed), so that it always remains in thermal equilibrium with the water (which has the same temperature everywhere). Which of the two bubbles is larger by the time they reach the surface? Explain your reasoning fully.

Problem 1.36. In the course of pumping up a bicycle tire, a liter of air at atmospheric pressure is compressed adiabatically to a pressure of 7 atm. (Air is mostly diatomic nitrogen and oxygen.)

(a) What is the final volume of this air after compression?

(b) How much work is done in compressing the air?

(c) If the temperature of the air is initially $300 \text{K}$ , what is the temperature after compression?

In Problem 1.16 you calculated the pressure of earth's atmosphere as a function of altitude, assuming constant temperature. Ordinarily, however, the temperature of the bottom most 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 |dT/dz| 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
$\frac{dT}{dP}=\frac{2}{f+2}\frac{T}{P}$
(b) Assume that dT/dz 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 1.16b to find a formula for dT/dz in this case. The result should be a constant, independent of temperature and pressure, which evaluates to approximately -10^oC/ km. This fundamental meteorological quantity is known as the dry adiabatic lapse rate.

By applying Newton’s laws to the oscillations of a continuous medium, one can show that the speed of a sound wave is given by

$c_s=\sqrt{\frac{B}{\rho }}$,

where $\rho$ is the density of the medium (mass per unit volume) and B is the bulk modulus, a measure of the medium’s stiffness? More precisely, if we imagine applying an increase in pressure $\mathrm{\Delta }P$ to a chunk of the material, and this increase results in a (negative) change in volume $\mathrm{\Delta }V$, then B is defined as the change in pressure divided by the magnitude of the fractional change in volume:

$B=\frac{\mathrm{\Delta }P}{–\mathrm{\Delta }V/V}$

This definition is still ambiguous, however, because I haven't said whether the compression is to take place isothermally or adiabatically (or in some other way).

1. Compute the bulk modulus of an ideal gas, in terms of its pressure P, for both isothermal and adiabatic compressions.
2. Argue that for purposes of computing the speed of a sound wave, the adiabatic B is the one we should use.
3. Derive an expression for the speed of sound in an ideal gas, in terms of its temperature and average molecular mass. Compare your result to the formula for the RMS speed of the molecules in the gas. Evaluate the speed of sound numerically for air at room temperature.
4. When Scotland’s Battlefield Band played in Utah, one musician remarked that the high altitude threw their bagpipes out of tune. Would you expect altitude to affect the speed of sound (and hence the frequencies of the standing waves in the pipes)? If so, in which direction? If not, why not?

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.

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?

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)?

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?

 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 $d T$ 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 $d P$ 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 $p$ ?