Question

(a) Use the van't Hoff equation in Problem 15.97 to derive the following expression, which relates the equilibrium constants at two different temperatures $$ \ln \frac{K_{1}}{K_{2}}=\frac{\Delta H^{\circ}}{R}\left(\frac{1}{T_{2}}-\frac{1}{T_{1}}\right) $$ How does this equation support the prediction based on Le Châtelier's principle about the shift in equilibrium with temperature? (b) The vapor pressures of water are \(31.82 \mathrm{mmHg}\) at \(30^{\circ} \mathrm{C}\) and \(92.51 \mathrm{mmHg}\) at \(50^{\circ} \mathrm{C} .\) Calculate the molar heat of vaporization of water.

Short answer

Deriving from the van't Hoff equation, we get \(\ln \frac{K_{1}}{K_{2}}=\frac{\Delta H^{\circ}}{R}\left(\frac{1}{T_{2}}-\frac{1}{T_{1}}\right)\), which effectively shows how equilibrium constant changes with temperature. This supports Le Châtelier's principle, predicting shifts in equilibrium with temperature changes. The molar heat of vaporization can be calculated using the Clausius–Clapeyron equation, given the vapor pressures at two temperature points.

Step-by-step solution

Understanding Van't Hoff Equation and Le Châtelier's Principle

The Van't Hoff equation demonstrates the temperature dependence of equilibrium constants. Le Châtelier's principle states that if a system in equilibrium suffers a disturbance (such as temperature and pressure changes), the system will adjust to partially counter that disturbance. In this context, if \(\Delta H^{\circ}\) is positive (meaning an endothermic reaction), the value of K (equilibrium constant) will increase as the temperature(T) rises to absorb excess heat, according to Le Châtelier's principle. Thus, the van’t Hoff equation gives a quantitative means of expressing this dependence.

Deriving the given expression

The Van 't Hoff equation is generally written as \(\frac {d(lnK)}{dT}=\frac {\Delta H^{\circ}}{RT^2}\). Integrating this between two temperatures \(T_{1}\) and \(T_{2}\) with corresponding equilibrium constants \(K_{1}\) and \(K_{2}\) would lead to the required expression.

Calculate the molar heat of vaporization

The Clausius–Clapeyron equation can be used to find the molar heat of vaporization (\(\Delta H^{\circ}\)). This is given as \(\ln \frac{P_{2}}{P_{1}}=-\frac{\Delta H^{\circ}}{R}\left(\frac{1}{T_{2}}-\frac{1}{T_{1}}\right)\), where \(P_{1}\) and \(P_{2}\) are the vapor pressures at temperatures \(T_{1}\) and \(T_{2}\). Plug in the given pressure and temperature values to find \(\Delta H^{\circ}\). Remember to convert the temperatures to Kelvin before substituting the values in to the formula.

Key concepts

Equilibrium Constants

Equilibrium constants are a way to express the ratio of product concentrations to reactant concentrations at equilibrium in a chemical reaction. They tell us about the position of equilibrium and how far the reaction will proceed. The symbol commonly used for equilibrium constants is \( K \).
If the value of \( K \) is large, it indicates a higher concentration of products compared to reactants, suggesting the reaction favors product formation. Conversely, a small \( K \) value suggests that reactants are more favored.

• Equilibrium constants allow us to predict the direction of a reaction under given conditions.
• They are dependent on temperature and can change if temperature conditions alter.

In the context of van’t Hoff equation, equilibrium constants help in understanding how equilibrium shifts with temperature changes.

Le Châtelier's Principle

Le Châtelier's principle is a fundamental concept in chemistry which states that a change in any condition that affects the equilibrium of a system will lead to a shift in the equilibrium position to counteract that change. This principle is vital in predicting how different factors like temperature, pressure, or concentration changes affect chemical systems.
According to this principle, if a system at equilibrium experiences an increase in temperature, it will shift to absorb the extra heat. For endothermic reactions, where the reaction takes in heat, increasing temperature will favor product formation.

• If you increase the concentration of a reactant, the system shifts to produce more products.
• Raising the temperature of an exothermic reaction shifts the equilibrium to favor reactants.

In relation to the van’t Hoff equation, Le Châtelier's principle helps explain why equilibrium constants change with temperature.

Molar Heat of Vaporization

Molar heat of vaporization refers to the amount of energy required to convert one mole of a liquid into vapor without a temperature change. This is a specific type of enthalpy change relevant to phase changes. For water, this represents the energy required to transition water from liquid to gas.
The Clausius–Clapeyron equation, \( \ln \frac{P_{2}}{P_{1}} = -\frac{\Delta H^{\circ}}{R}\left(\frac{1}{T_{2}}-\frac{1}{T_{1}}\right) \), allows us to calculate the molar heat of vaporization when given two pressures and two temperatures.

• This equation exemplifies how vapor pressure changes with temperature.
• Useful in engineering and the sciences whenever phase changes are involved.

The van’t Hoff equation provides a similar approach but focuses on reactions and equilibrium constants rather than phase changes.

Temperature Dependence

Temperature dependence in chemistry explains how chemical reaction rates and equilibria are affected by changes in temperature. Many reactions speed up with temperature increases due to higher kinetic energy, which increases the chances of reactant collisions.
For equilibrium constants, temperature dependence is crucial. The van't Hoff equation, \( \frac {d(\ln K)}{dT}=\frac {\Delta H^{\circ}}{RT^2} \), provides a mathematical basis for this dependence, showing that changes in temperature directly influence \( K \) through the enthalpy \( \Delta H^{\circ} \).

• Temperature influences equilibrium constants. Higher temperatures can increase \( K \) for endothermic reactions and decrease \( K \) for exothermic reactions.
• The rate constants within the rate laws are also temperature-dependent, enhancing the reaction's speed as temperature rises.

Understanding temperature dependence is vital for predicting the behavior of chemical reactions under different thermal conditions.