Question

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 $2.3 \times 10^6 \mathrm{~J} \mathrm{~kg}^{-1}$.

Name: Problem 5, Chapter 1: Statistical Physics of Particles
Uploaded: 2020-09-28T14:55:47-08:00
Duration: 5 min 53 s
Channel: Lottie Adams

   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 $2.3 \times 10^6 \mathrm{~J} \mathrm{~kg}^{-1}$.

Statistical Physics of Particles

Mehran Kardar 1st Edition

Chapter 1, Problem 5

Instant Answer

Solved by Expert Lottie Adams

09/28/2020

Step 1

- When water is converted to steam at pressure $ P $ and temperature $ T $, the volume increases by $ V $. The pressure is then adiabatically decreased to $ P - \mathrm{d}P $. - The work done by the system during this expansion is $ W = P \Delta V $. Show more…

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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 $2.3 \times 10^6 \mathrm{~J} \mathrm{~kg}^{-1}$.

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Key Concepts

Clausius-Clapeyron Equation

This equation relates the slope of the phase boundary (dP/dT) to the latent heat (L) and the volume change (V) during a phase transition. It is derived by considering the equilibrium condition between two phases (like liquid and gas) and shows that the rate at which the boiling point changes with pressure is proportional to the energy required for the phase change per unit of temperature and volume change.

Carnot Cycle and Thermodynamic Efficiency

The Carnot cycle is an idealized thermodynamic cycle that affords the maximum possible efficiency for converting heat into work, based solely on the temperatures of the heat source and sink. Its principles, such as the extraction of work from heat transfer and the definition of reversible processes, underlie the derivation of many thermodynamic relationships, including alternative derivations of the Clausius-Clapeyron equation by quantifying the work output in phase transitions.

Latent Heat in Phase Transitions

Latent heat is the heat absorbed or released during a phase change at constant temperature and pressure. This energy, which does not change the temperature of the substance, is crucial in processes like boiling, where it accounts for the energy required to overcome molecular interactions. In derivations such as the Clausius-Clapeyron equation, latent heat links the energy exchange to changes in pressure and volume during the phase transition.

Ideal Gas Behavior

When one of the phases, such as steam, is well-approximated as an ideal gas, its properties (like volume, pressure, and temperature) are connected via the ideal gas law. This approximation is used to simplify relationships in problems involving phase transitions and is particularly useful when integrating equations like the Clausius-Clapeyron equation to obtain expressions for pressure as a function of temperature.

Barometric Formula

The barometric formula describes how atmospheric pressure decreases exponentially with altitude due to the weight of the overlying air. By balancing the gravitational forces with the pressure gradient in a column of air treated as an ideal gas, this formula provides a fundamental link between pressure, altitude, and temperature in atmospheric physics. It plays an important role when analyzing environmental effects on phase transitions, such as variations in the boiling point of water at high altitudes.

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