Question

The vapor pressure of liquid \(\mathrm{X}\) is lower than that of liquid \(Y\) at \(20^{\circ} \mathrm{C}\), but higher at \(60^{\circ} \mathrm{C}\). What can you deduce about the relative magnitude of the molar heats of vaporization of \(\mathrm{X}\) and \(\mathrm{Y}\) ?

Short answer

Liquid X has a higher molar heat of vaporization than liquid Y.

Step-by-step solution

Understanding Vapor Pressure

Vapor pressure is the pressure exerted by a vapor in thermodynamic equilibrium with its condensed phases at a given temperature in a closed system. A higher vapor pressure at a given temperature means the substance is more volatile and more molecules are in the vapor phase.

Analyzing Given Information at 20°C

At 20°C, the vapor pressure of liquid X is lower than that of liquid Y, indicating that at this temperature, liquid Y is more volatile and transforms into vapor more easily than liquid X.

Analyzing Given Information at 60°C

At 60°C, the vapor pressure of liquid X is higher than that of liquid Y. This indicates a change in the volatility relationship between X and Y at different temperatures.

Understanding Clausius-Clapeyron Relation

The Clausius-Clapeyron equation relates the vapor pressure and temperature, showing that the vapor pressure of a liquid increases exponentially with temperature. The equation is \( \ln \left( \frac{P_2}{P_1} \right) = \frac{\Delta H_{vap}}{R} \left( \frac{1}{T_1} - \frac{1}{T_2} \right) \), where \( \Delta H_{vap} \) is the molar heat of vaporization and R is the gas constant.

Comparing Heats of Vaporization

Since liquid X's vapor pressure increases more dramatically compared to Y as temperature rises from 20°C to 60°C, liquid X likely has a higher molar heat of vaporization than liquid Y. A higher \( \Delta H_{vap} \) indicates a larger increase in vapor pressure with temperature.

Key concepts

Molar Heat of Vaporization

The molar heat of vaporization, denoted as \( \Delta H_{vap} \), refers to the amount of energy needed to vaporize one mole of a substance. This crucial thermodynamic property indicates how much heat is required to turn a liquid into vapor without a change in temperature.

It plays a vital role in understanding how substances behave under heat and pressure changes. Generally speaking, a higher \( \Delta H_{vap} \) suggests that a substance requires more energy to escape into the vapor phase. Therefore, this means it may be less volatile at lower temperatures.

This concept ties directly into how substances interact with temperature increases, such as the shift observed between the 20°C and 60°C conditions for liquids X and Y in the problem.

Clausius-Clapeyron Equation

The Clausius-Clapeyron equation is a fundamental principle that connects vapor pressure and temperature. It provides a mathematical way to express how vapor pressure changes with temperature and can be written as:\[ \ln \left( \frac{P_2}{P_1} \right) = \frac{\Delta H_{vap}}{R} \left( \frac{1}{T_1} - \frac{1}{T_2} \right) \]
The variables in the equation include:

• \( \Delta H_{vap} \) - Molar heat of vaporization
• \( R \) - Universal gas constant
• \( T_1 \) and \( T_2 \) - Temperatures in Kelvin
• \( P_1 \) and \( P_2 \) - Vapor pressures at \( T_1 \) and \( T_2 \) respectively

This equation is particularly helpful for calculating the vapor pressures at different temperatures if \( \Delta H_{vap} \) is known and vice versa. In our problem, liquid X's vapor pressure increases more dramatically than that of liquid Y as temperature rises, supporting the idea that its \( \Delta H_{vap} \) is higher.

Thermodynamic Equilibrium

Thermodynamic equilibrium is a state in which all thermodynamic processes are in a balance, and macroscopic quantities like temperature and pressure remain constant over time. In simpler terms, it's when a system has no net change occurring. For vapor pressure, it signifies that the rate at which molecules are evaporating equals the rate at which they condense back into the liquid.

This concept is crucial because vapor pressure is defined under these conditions, allowing us to study the system without interference from external changes.

In the problem's context, understanding thermodynamic equilibrium helps explain why changes in vapor pressures signify differences in other properties, like the molar heat of vaporization of the substances in question.

Volatility

Volatility refers to the ability of a substance to vaporize. The more volatile a substance, the more easily it transforms from liquid to vapor under a particular temperature. Substances with high vapor pressures at given temperatures are considered to be more volatile.
Factors affecting volatility include:

• Intermolecular forces: Weaker forces increase volatility as less energy is needed to break these bonds.
• Molar heat of vaporization: Lower \( \Delta H_{vap} \) often means higher volatility since less energy is required for vaporization.

The comparison between liquids X and Y at different temperatures provides a practical example of changing volatility. At 60°C, liquid X becomes more volatile than liquid Y, indicating stronger temperature effects on its vapor pressure.