Chapter 1, Thermodynamics Video Solutions, Statistical Physics of Particles

Problem 1

Surface tension: thermodynamic properties of the interface between two phases are described by a state function called the surface tension $\delta$. It is defined in terms of the work required to increase the surface area by an amount $\mathrm{d} A$ through $\mathrm{d} W=\mathcal{} \mathrm{d} A$.
(a) By considering the work done against surface tension in an infinitesimal change in radius, show that the pressure inside a spherical drop of water of radius $R$ is larger than outside pressure by $2 \mathcal{S} / R$. What is the air pressure inside a soap bubble of radius $R$ ?
(b) A water droplet condenses on a solid surface. There are three surface tensions involved, $\mathcal{S}_{a w}, \mathcal{S}_{s w}$, and $\mathcal{S}_{s a}$, where $a, s$, and $w$ refer to air, solid, and water, respectively. Calculate the angle of contact, and find the condition for the appearance of a water film (complete wetting).
(c) In the realm of "large" bodies gravity is the dominant force, while at "small" distances surface tension effects are all important. At room temperature, the surface tension of water is $\mathcal{S}_o \approx 7 \times 10^{-2} \mathrm{~N} \mathrm{~m}^{-1}$. Estimate the typical length scale that separates "large" and "small" behaviors. Give a couple of examples for where this length scale is important.

Ryan Hood

Numerade Educator

01:14

Problem 2

Surfactants: surfactant molecules such as those in soap or shampoo prefer to spread on the air-water surface rather than dissolve in water. To see this, float a hair on the surface of water and gently touch the water in its vicinity with a piece of soap. (This is also why a piece of soap can power a toy paper boat.)
(a) The air-water surface tension $\mathcal{S}_o$ (assumed to be temperature-independent) is reduced roughly by $N k_B T / A$, where $N$ is the number of surfactant particles, and $A$ is the area. Explain this result qualitatively.
(b) Place a drop of water on a clean surface. Observe what happens to the air-water surface contact angle as you gently touch the droplet surface with a small piece of soap, and explain the observation.
(c) More careful observations show that at higher surfactant densities
$$
\left.\frac{\partial \mathcal{S}}{\partial A}\right|_T=\frac{N k_B T}{(A-N b)^2}-\frac{2 a}{A}\left(\frac{N}{A}\right)^2 \quad, \quad \text { and }\left.\quad \frac{\partial T}{\partial \mathcal{S}}\right|_A=-\frac{A-N b}{N k_B},
$$
where $a$ and $b$ are constants. Obtain the expression for $\mathcal{S}(A, T)$ and explain qualitatively the origin of the corrections described by $a$ and $b$.
(d) Find an expression for $C_B-C_A$ in terms of $\left.\frac{\partial E}{\partial A}\right|_T=\left.\frac{\partial E}{\partial A}\right|_\delta, \mathcal{S}$, $\left.\frac{\partial \delta}{\partial A}\right|_T$, and $\left.\frac{\partial T}{\partial S}\right|_A$.
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Sana Riaz

Numerade Educator

01:43

Problem 3

Temperature scales: prove the equivalence of the ideal gas temperature scale $\Theta$, and the thermodynamic scale $T$, by performing a Carnot cycle on an ideal gas. The ideal gas satisfies $P V=N k_B \Theta$, and its internal energy $E$ is a function of $\Theta$ only. However, you may not assume that $E \propto \Theta$. You may wish to proceed as follows:
(a) Calculate the heat exchanges $Q_H$ and $Q_C$ as a function of $\Theta_H, \Theta_C$, and the volume expansion factors.
(b) Calculate the volume expansion factor in an adiabatic process as a function of $\Theta$.
(c) Show that $Q_H / Q_C=\Theta_H / \Theta_C$.

Penny Riley

Numerade Educator

01:00

Problem 4

Equations of state: the equation of state constrains the form of internal energy as in the following examples.
(a) Starting from $\mathrm{d} E=T \mathrm{~d} S-P \mathrm{~d} V$, show that the equation of state $P V=N k_B T$ in fact implies that $E$ can only depend on $T$.
(b) What is the most general equation of state consistent with an internal energy that depends only on temperature?
(c) Show that for a van der Waals gas $C_V$ is a function of temperature alone.

Mishal Gul

Numerade Educator

05:53

Problem 5

The Clausius-Clapeyron equation describes the variation of boiling point with pressure. It is usually derived from the condition that the chemical potentials of the gas and liquid phases are the same at coexistence. For an alternative derivation, consider a Carnot engine using one mole of water. At the source $(P, T)$ the latent heat $L$ is supplied converting water to steam. There is a volume increase $V$ associated with this process. The pressure is adiabatically decreased to $P-\mathrm{d} P$. At the $\operatorname{sink}(P-\mathrm{d} P, T-\mathrm{d} T)$ steam is condensed back to water.
(a) Show that the work output of the engine is $W=V \mathrm{~d} P+\mathcal{O}\left(\mathrm{d} P^2\right)$. Hence obtain the Clausius-Clapeyron equation
$$
\left.\frac{\mathrm{d} P}{\mathrm{~d} T}\right|_{\text {boiling }}=\frac{L}{T V} .
$$
(b) What is wrong with the following argument: "The heat $Q_H$ supplied at the source to convert one mole of water to steam is $L(T)$. At the sink $L(T-\mathrm{d} T)$ is supplied to condense one mole of steam to water. The difference $\mathrm{d} T \mathrm{~d} L / \mathrm{d} T$ must equal the work $W=V \mathrm{~d} P$, equal to $L \mathrm{~d} T / T$ from Eq. (1). Hence $\mathrm{d} L / \mathrm{d} T=L / T$, implying that $L$ is proportional to $T$ !"
(c) Assume that $L$ is approximately temperature-independent, and that the volume change is dominated by the volume of steam treated as an ideal gas, that is, $V=N k_B T / P$. Integrate Eq. (1) to obtain $P(T)$.
(d) A hurricane works somewhat like the engine described above. Water evaporates at the warm surface of the ocean, steam rises up in the atmosphere, and condenses to water at the higher and cooler altitudes. The Coriolis force converts the upward suction of the air to spiral motion. (Using ice and boiling water, you can create a little storm in a tea cup.) Typical values of warm ocean surface and high altitude temperatures are $80^{\circ} \mathrm{F}$ and $-120^{\circ} \mathrm{F}$, respectively. The warm water surface layer must be at least 200 feet thick to provide sufficient water vapor, as the hurricane needs to condense about 90 million tons of water vapor per hour to maintain itself. Estimate the maximum possible efficiency, and power output, of such a hurricane. (The latent heat of vaporization of water is about $2.3 \times 10^6 \mathrm{~J} \mathrm{~kg}^{-1}$.)
(e) Due to gravity, atmospheric pressure $P(h)$ drops with the height $h$. By balancing the forces acting on a slab of air (behaving like a perfect gas) of thickness $\mathrm{d} h$, show that $P(h)=P_0 \exp (-m g h / k T)$, where $m$ is the average mass of a molecule in air.
(f) Use the above results to estimate the boiling temperature of water on top of Mount Everest $(h \approx 9 \mathrm{~km})$. The latent heat of vaporization of water is about $2.3 \times 10^6 \mathrm{~J} \mathrm{~kg}^{-1}$.

Lottie Adams

Numerade Educator

04:03

Problem 6

Glass: liquid quartz, if cooled slowly, crystallizes at a temperature $T_m$, and releases latent heat $L$. Under more rapid cooling conditions, the liquid is supercooled and becomes glassy.
(a) As both phases of quartz are almost incompressible, there is no work input, and changes in internal energy satisfy $\mathrm{d} E=T \mathrm{~d} S+\mu \mathrm{d} N$. Use the extensivity condition to obtain the expression for $\mu$ in terms of $E, T, S$, and $N$.
(b) The heat capacity of crystalline quartz is approximately $C_X=\alpha T^3$, while that of glassy quartz is roughly $C_G=\beta T$, where $\alpha$ and $\beta$ are constants.
Assuming that the third law of thermodynamics applies to both crystalline and glass phases, calculate the entropies of the two phases at temperatures $T \leq T_m$.
(c) At zero temperature the local bonding structure is similar in glass and crystalline quartz, so that they have approximately the same internal energy $E_0$. Calculate the internal energies of both phases at temperatures $T \leq T_m$
(d) Use the condition of thermal equilibrium between two phases to compute the equilibrium melting temperature $T_m$ in terms of $\alpha$ and $\beta$.
(e) Compute the latent heat $L$ in terms of $\alpha$ and $\beta$.
(f) Is the result in the previous part correct? If not, which of the steps leading to it is most likely to be incorrect?

Ankur S

Numerade Educator

03:55

Problem 7

Filament: for an elastic filament it is found that, at a finite range in temperature, a displacement $x$ requires a force
$$
J=a x-b T+c T x
$$
where $a, b$, and $c$ are constants. Furthermore, its heat capacity at constant displacement is proportional to temperature, that is, $C_x=A(x) T$.
(a) Use an appropriate Maxwell relation to calculate $\partial S /\left.\partial x\right|_T$.
(b) Show that $A$ has to in fact be independent of $x$, that is, $\mathrm{d} A / \mathrm{d} x=0$.
(c) Give the expression for $S(T, x)$ assuming $S(0,0)=S_0$.
(d) Calculate the heat capacity at constant tension, that is, $C_J=T \partial S /\left.\partial T\right|_J$ as a function of $T$ and $J$.
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Mukesh Devi

Numerade Educator

02:37

Problem 8

Hard core gas: a gas obeys the equation of state $P(V-N b)=N k_B T$, and has a heat capacity $C_V$ independent of temperature. ( $N$ is kept fixed in the following.)
(a) Find the Maxwell relation involving $\partial S /\left.\partial V\right|_{T, N}$.
(b) By calculating $\mathrm{d} E(T, V)$, show that $E$ is a function of $T$ (and $N$ ) only.
(c) Show that $\gamma \equiv C_P / C_V=1+N k_B / C_V$ (independent of $T$ and $V$ ).
(d) By writing an expression for $E(P, V)$, or otherwise, show that an adiabatic change satisfies the equation $P(V-N b)^\gamma=$ constant.

Pritesh Ranjan

Numerade Educator

11:14

Problem 9

Superconducting transition: many metals become superconductors at low temperatures $T$, and magnetic fields $B$. The heat capacities of the two phases at zero magnetic field are approximately given by
$$
\left\{\begin{array}{lc}
C_s(T)=V \alpha T^3 & \text { in the superconducting phase } \\
C_n(T)=V\left[\beta T^3+\gamma T\right] & \text { in the normal phase }
\end{array},\right.
$$
where $V$ is the volume, and $\{\alpha, \beta, \gamma\}$ are constants. (There is no appreciable change in volume at this transition, and mechanical work can be ignored throughout this problem.)
(a) Calculate the entropies $S_s(T)$ and $S_n(T)$ of the two phases at zero field, using the third law of thermodynamics.
(b) Experiments indicate that there is no latent heat $(L=0)$ for the transition between the normal and superconducting phases at zero field. Use this information to obtain the transition temperature $T_c$, as a function of $\alpha, \beta$, and $\gamma$.
(c) At zero temperature, the electrons in the superconductor form bound Cooper pairs. As a result, the internal energy of the superconductor is reduced by an amount $V \Delta$, that is, $E_n(T=0)=E_0$ and $E_s(T=0)=E_0-V \Delta$ for the metal and superconductor, respectively. Calculate the internal energies of both phases at finite temperatures.
(d) By comparing the Gibbs free energies (or chemical potentials) in the two phases, obtain an expression for the energy gap $\Delta$ in terms of $\alpha, \beta$, and $\gamma$.
(e) In the presence of a magnetic field $B$, inclusion of magnetic work results in $\mathrm{d} E=$ $T \mathrm{~d} S+B \mathrm{~d} M+\mu \mathrm{d} N$, where $M$ is the magnetization. The superconducting phase is a perfect diamagnet, expelling the magnetic field from its interior, such that $M_s=$ $-V B /(4 \pi)$ in appropriate units. The normal metal can be regarded as approximately non-magnetic, with $M_n=0$. Use this information, in conjunction with previous results, to show that the superconducting phase becomes normal for magnetic fields larger than
$$
B_c(T)=B_0\left(1-\frac{T^2}{T_c^2}\right),
$$
giving an expression for $B_0$.

Khoobchandra Agrawal

Numerade Educator

05:15

Problem 10

Photon gas Carnot cycle: the aim of this problem is to obtain the black-body radiation relation, $E(T, V) \propto V T^4$, starting from the equation of state, by performing an infinitesimal Carnot cycle on the photon gas.
(a) Express the work done, $W$, in the above cycle, in terms of $\mathrm{d} V$ and $\mathrm{d} P$.
(b) Express the heat absorbed, $Q$, in expanding the gas along an isotherm, in terms of $P, \mathrm{~d} V$, and an appropriate derivative of $E(T, V)$.
(c) Using the efficiency of the Carnot cycle, relate the above expressions for $W$ and $Q$ to $T$ and $\mathrm{d} T$.
(d) Observations indicate that the pressure of the photon gas is given by $P=A T^4$, where $A=\pi^2 k_B^4 / 45(\hbar c)^3$ is a constant. Use this information to obtain $E(T, V)$, assuming $E(0, V)=0$.
(e) Find the relation describing the adiabatic paths in the above cycle.

Salamat Ali

Numerade Educator

01:45

Problem 11

Irreversible processes
(a) Consider two substances, initially at temperatures $T_1^0$ and $T_2^0$, coming to equilibrium at a final temperature $T_f$ through heat exchange. By relating the direction of heat flow to the temperature difference, show that the change in the total entropy, which can be written as
$$
\Delta S=\Delta S_1+\Delta S_2 \geq \int_{T_1^0}^{T_f} \frac{\mathrm{d} Q_1}{T_1}+\int_{T_2^0}^{T_f} \frac{\mathrm{d} Q_2}{T_2}=\int \frac{T_1-T_2}{T_1 T_2} \mathrm{~d} Q,
$$
must be positive. This is an example of the more general condition that "in a closed system, equilibrium is characterized by the maximum value of entropy $S . "$
(b) Now consider a gas with adjustable volume $V$, and diathermal walls, embedded in a heat bath of constant temperature $T$, and fixed pressure $P$. The change in the entropy of the bath is given by
$$
\Delta S_{\text {bath }}=\frac{\Delta Q_{\text {bath }}}{T}=-\frac{\Delta Q_{\text {gas }}}{T}=-\frac{1}{T}\left(\Delta E_{\text {gas }}+P \Delta V_{\mathrm{gas}}\right) .
$$
By considering the change in entropy of the combined system establish that "the equilibrium of a gas at fixed $T$ and $P$ is characterized by the minimum of the Gibbs free energy $G=E+P V-T S . "$

Khoobchandra Agrawal

Numerade Educator

01:11

Problem 12

The Solar System originated from a dilute gas of particles, sufficiently separated from other such clouds to be regarded as an isolated system. Under the action of gravity the particles coalesced to form the Sun and planets.
(a) The motion and organization of planets is much more ordered than the original dust cloud. Why does this not violate the second law of thermodynamics?
(b) The nuclear processes of the Sun convert protons to heavier elements such as carbon. Does this further organization lead to a reduction in entropy?
(c) The evolution of life and intelligence requires even further levels of organization. How is this achieved on Earth without violating the second law?

Raj Bala

Numerade Educator