Question 1

Which equation describes the slope of a two-phase equilibrium line on a one-component phase diagram, and what does each term represent in the equation $$\frac{dp}{dT} = \frac{\Delta H_t}{T \Delta V_t}$$?

Accepted Answer

The Clausius-Clapeyron equation; $$\Delta H_t is $$the enthalpy change of transition, $$T is $$temperature, and $$\Delta V_t is $$the change in molar volume.

Question 2

A sample of ice and water is at equilibrium at 273 K. If the pressure is increased, what happens to the melting point of ice? Use the Clapeyron equation to justify your answer.

Accepted Answer

The melting point decreases because $$\Delta V_{fus}$$ for water is negative, so $$\frac{dT}{dp} is $$negative.

Question 3

Which of the following is a key assumption made in deriving the Clausius-Clapeyron equation for liquid-vapour equilibrium?

Accepted Answer

The molar volume of the vapour is much greater than that of the liquid.

Question 4

Given the integrated Clausius-Clapeyron equation $$\ln \left( \frac{p_2$}{p_1$} \right) = -\frac{\Delta H_{vap}}{R} \left( \frac{1}{T_2$} - \frac{1}{T_1$} \right)$$, what is the physical meaning of the slope when plotting $$\ln(p)$$ versus $$1/T$$?

Accepted Answer

The slope is $$-\frac{\Delta H_{vap}}{R}$$, which gives the molar enthalpy of vaporization.

Question 5

A liquid has a normal boiling point of $$-93^\circC $$and a vapour pressure of 0.296 atm at $$-113^\circC. $$Estimate the molar enthalpy of vaporization using the Clausius-Clapeyron equation. (Use $$R = 8.314 J $$mol$$^{-1} K$^{-1})$$

Accepted Answer

$$\Delta H_{vap} = 14.6 kJ $$mol$$^{-1}$$

Question 6

Why is the solid–liquid equilibrium line for water sloped to the left in a phase diagram, unlike most other substances?

Accepted Answer

Because the molar volume of ice is greater than that of liquid water, so $$\Delta V_{fus} is $$negative.

Question 7

Which of the following best explains why the solid–liquid equilibrium line is steeper than the solid–vapour or liquid–vapour lines in a phase diagram?

Accepted Answer

Melting involves a much smaller change in volume than vaporization or sublimation, resulting in a larger slope according to $$\frac{dp}{dT} = \frac{\Delta H}{T \Delta V}.$$

Question 8

Which of the following statements about metastable phases is correct?

Accepted Answer

A metastable phase exists when its Gibbs free energy is higher than that of the stable phase, but the rate of conversion is slow.

Question 9

Sulfur can exist in more than one solid phase. What is the term for this phenomenon, and what is the term for the same phenomenon in elements?

Accepted Answer

Polymorphism; allotropy

Question 10

Which of the following is NOT an assumption made in the derivation of the Clausius-Clapeyron equation for vaporization?

Accepted Answer

The vapour behaves as a non-ideal gas.

Question 11

If the enthalpy of melting of ice is $$6.03 kJ $$mol$$^{-1}$$, the molar volumes of ice and water at 273 K are $$19.6 cm$^3$$ mol$$^{-1}$$ and $$18.0 cm$^3$$ mol$$^{-1}$$ respectively, calculate $$\frac{dT}{dp}$$ for the melting point of ice in K bar$$^{-1}. (1 $$bar = $$10^5 Pa)$$

Accepted Answer

$$-0.0072 K $$bar$$^{-1}$$

Question 12

Which of the following best describes the physical meaning of the Clapeyron equation?

Accepted Answer

It relates the slope of a phase boundary to the enthalpy and volume changes during a phase transition.

Question 13

In the context of the Clausius-Clapeyron equation, why is it important that $$\Delta H is $$approximately constant over a small temperature range?

Accepted Answer

It allows the equation to be integrated easily, providing a linear relationship between $$\ln(p)$$ and $$1/T.$$

Question 14

Which of the following real-world examples involves a metastable phase?

Accepted Answer

Diamond at 25°C and 1 atm

Question 15

Which of the following is a correct integrated form of the Clausius-Clapeyron equation for vaporization?

Accepted Answer

$$\ln \left( \frac{p_2$}{p_1$} \right) = -\frac{\Delta H_{vap}}{R} \left( \frac{1}{T_2$} - \frac{1}{T_1$} \right)$$

The Clapeyron and Clausius-Clapeyron Equations: Thermodynamics of Phase Transitions

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