11.9
Clausius-Clapeyron Equation
The equilibrium between a liquid and its vapor depends on the temperature of the system; a rise in temperature causes a corresponding rise in the vapo…
Recall that the vapor pressure of a liquid increases with a rise in temperature. However, this dependence is not linear.
To illustrate, the vapor pressure of water at 50 °C is 0.122 atm, while at 100 °C, it is 1 atm. Vapor pressure curves sharply upward with increasing temperature, resulting in an exponential curve.
In comparison, when the natural log of vapor pressure is plotted against reciprocal temperature, a straight line is obtained, and its equation is called the Clausius–Clapeyron equation.
Here, R is the ideal gas constant; c, a constant characteristic of the liquid, is the y-intercept; and the slope of the line is equal to the negative of the molar heat of vaporization over the gas constant.
The equation allows the calculation of the molar heat of vaporization from the experimental measurements of equilibrium vapor pressures and temperatures.
For example, suppose the natural log of ethanol vapor pressure plotted as a function of reciprocal temperature gives a straight line with the slope of negative 4638 kelvins.
The equation for the slope of the line, along with the value of R, gives the molar heat of vaporization of ethanol as 38 560 joules per mole.
If the molar heat of vaporization of any liquid and its vapor pressure at one temperature are known, the equation’s two-point form can be used to calculate the liquid’s vapor pressure at a different temperature.
Take the example of water, whose enthalpy of vaporization is 40.7 kilojoules per mole. If the vapor pressure of water at 373 kelvins is 1 atm, what will be its vapor pressure at 383 kelvins?
To solve, use the two-point form of the equation and substitute the given values of vapor pressure, enthalpy of vaporization, the two temperatures, and the gas constant to get the vapor pressure of water at 383 kelvins as 1.409 atm.
The rise in vapor pressure from 373 kelvins to 383 kelvins is 0.409 atm, which clearly indicates that an increase in vapor pressure as a function of temperature is a non-linear process.
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Q1: Why is the relationship between vapor pressure and temperature not linear?
Vapor pressure increases exponentially with temperature rather than in a straight line. When vapor pressure is plotted directly against temperature, the curve rises sharply upward. However, when the natural logarithm of vapor pressure is plotted against reciprocal temperature, a linear relationship emerges, which is described by the Clausius-Clapeyron equation.
Q2: What does the slope of a Clausius-Clapeyron plot represent?
The slope of the line in a Clausius-Clapeyron plot equals the negative molar heat of vaporization divided by the gas constant. For example, if a plot of ethanol vapor pressure yields a slope of negative 4638 kelvins, you can calculate the molar heat of vaporization by multiplying the slope's absolute value by the gas constant, yielding 38,560 joules per mole.
Q3: How can you predict vapor pressure at a different temperature using the Clausius-Clapeyron equation?
If you know the molar heat of vaporization and vapor pressure at one temperature, use the two-point form of the Clausius-Clapeyron equation to calculate vapor pressure at another temperature. For water with enthalpy of vaporization of 40.7 kilojoules per mole and vapor pressure of 1 atm at 373 kelvins, the equation predicts 1.409 atm at 383 kelvins, demonstrating the non-linear increase in vapor pressure.
Q4: What is the constant A in the Clausius-Clapeyron equation?
The constant A is the y-intercept of the Clausius-Clapeyron plot and is characteristic of each liquid. Its value depends on the chemical identity of the substance. While A remains constant for a given liquid across different temperatures, it varies between different liquids based on their molecular properties and intermolecular interactions.
Q5: Why must temperature be expressed in kelvins in the Clausius-Clapeyron equation?
The Clausius-Clapeyron equation uses absolute temperature in kelvins because the relationship between vapor pressure and temperature is fundamentally based on thermodynamic principles involving the gas constant R. Using kelvins ensures the mathematical relationships and proportionalities in the equation remain valid across all temperature ranges.
Q6: What information do you need to use the two-point form of the Clausius-Clapeyron equation?
To use the two-point form, you need the molar heat of vaporization, vapor pressures at two different temperatures, those two temperatures in kelvins, and the gas constant. With these values, you can rearrange the equation to solve for an unknown vapor pressure at a new temperature or verify the relationship between vapor pressure and temperature for any liquid.
Q7: How does the Clausius-Clapeyron equation relate to the equilibrium between a liquid and its vapor?
The Clausius-Clapeyron equation quantifies how vapor pressure—the pressure at which a liquid and its vapor are in equilibrium—changes with temperature. It predicts the rate at which vapor pressure increases per unit temperature increase, allowing chemists to understand and calculate the equilibrium conditions between liquid and vapor phases at different temperatures.