Question

If the vapor pressure of \(\mathrm{CC} 1_{4}\) (carbon tetrachloride) is \(.132\) atm at \(23^{\circ} \mathrm{C}\) and \(.526 \mathrm{~atm}\) at \(58^{\circ} \mathrm{C}\), what is the \(\Delta \mathrm{H}^{\prime}\) in this temperature range?

Short answer

The enthalpy change \(\Delta \mathrm{H}^{\prime}\) in the given temperature range for carbon tetrachloride (\(\mathrm{CCl_4}\)) is approximately \(-112460 \;\mathrm{J}\cdot\mathrm{mol}^{-1}\).

Step-by-step solution

Write down the Clausius-Clapeyron equation

The Clausius-Clapeyron equation is given as: \[ \ln\left(\frac{P_2}{P_1}\right) = \frac{\Delta \mathrm{H}^{\prime}}{R} \left(\frac{1}{T_1}-\frac{1}{T_2}\right) \] where: \(P_1\) and \(P_2\) are the vapor pressures at temperatures \(T_1\) and \(T_2\) respectively, \(\Delta \mathrm{H}^{\prime}\) is the enthalpy change, \(R\) is the ideal gas constant, \(8.314 \mathrm{J}\cdot\mathrm{mol}^{-1}\cdot\mathrm{K}^{-1}\).

Plug in the given values

Given vapor pressures and temperatures are: \(P_1 = 0.132 \;\text{atm}\) \(P_2 = 0.526 \;\text{atm}\) \(T_1 = (23 + 273.15) \;\text{K} = 296.15 \;\text{K}\) \(T_2 = (58 + 273.15) \;\text{K} = 331.15 \;\text{K}\) Plug these values into the Clausius-Clapeyron equation: \( \ln\left(\frac{0.526}{0.132}\right) = \frac{\Delta \mathrm{H}^{\prime}}{8.314} \left(\frac{1}{296.15}-\frac{1}{331.15}\right) \)

Solve for \(\Delta \mathrm{H}^{\prime}\)

Now, solve the equation for \(\Delta \mathrm{H}^{\prime}\): 1. Calculate the logarithm: \( \ln\left(\frac{0.526}{0.132}\right) \approx 1.655 \) 2. Calculate the temperature difference: \(\left(\frac{1}{296.15}-\frac{1}{331.15}\right) \approx -0.0001218\) Now plug these values back into the equation and solve for \(\Delta \mathrm{H}^{\prime}\): \( 1.655 = \frac{\Delta \mathrm{H}^{\prime}}{8.314} \times -0.0001218 \) \( \Delta \mathrm{H}^{\prime} = 1.655 \times 8.314 \div -0.0001218 \approx -112460 \;\mathrm{J}\cdot\mathrm{mol}^{-1} \) Therefore, the enthalpy change \(\Delta \mathrm{H}^{\prime}\) in the given temperature range is approximately \(-112460 \;\mathrm{J}\cdot\mathrm{mol}^{-1}\).

Key concepts

Vapor Pressure

Vapor pressure is a measure of the tendency of a substance to vaporize. It is defined as the pressure exerted by the vapor in equilibrium with its liquid (or solid) phase at a given temperature in a closed system. The vapor pressure of a liquid increases with temperature, as more molecules have sufficient kinetic energy to escape the surface and enter the vapor phase.

When considering the vapor pressure of substances, it's important to recognize that it's a crucial factor in understanding boiling point, evaporation, and other phase-transition phenomena. For example, when the vapor pressure equals atmospheric pressure, the liquid will boil. This understanding helps explain why the Clapeyron-Clausius equation, which relates vapor pressure and temperature, is an essential tool in physical chemistry and engineering.

Enthalpy Change

Enthalpy change, denoted by \( \Delta H \) and also referred to as heat of reaction, represents the total heat content change of a system at constant pressure during a chemical reaction or a phase transition. This thermodynamic quantity is a crucial measure for understanding energetics in physical and chemical processes.

Revealing the heat absorbed or released, the enthalpy change serves as a predictor for reaction spontaneity and favorability under specific conditions. In endothermic processes, where heat is absorbed, \( \Delta H \) is positive; conversely, it's negative for exothermic processes. The Clausius-Clapeyron equation not only ties vapor pressure with temperature but also encapsulates the enthalpy change of the phase transition from a liquid to a vapor.

Ideal Gas Constant

The ideal gas constant, represented as \( R \) in equations, plays a pivotal role in thermodynamics and physical chemistry. Its value is approximately 8.314 J\(\cdot\)mol\(^{-1}\)\(\cdot\)K\(^{-1}\) and serves as a bridge binding the macroscopic world to molecular-level phenomena.

In the context of the Clausius-Clapeyron equation, \( R \) helps to quantify the energy terms associated with the phase transitions by relating pressure, temperature, and the amount of substance in moles. This constant is also featured in the universal gas equation and embodies the relationship between particle kinetics and thermodynamics in various applications, ranging from engine efficiency calculations to the study of atmospheric conditions.

Temperature Dependence of Vapor Pressure

The relationship between temperature and vapor pressure is logarithmic and nonlinear, implying that a substance's vapor pressure will not increase at a constant rate with temperature. According to the Clausius-Clapeyron equation, a straight line is obtained when plotting the natural logarithm of vapor pressure against the inverse of temperature, showcasing the exponential nature of the relationship.

The equation reveals that for many substances, slight temperature variations at lower temperatures result in smaller changes in vapor pressure than at higher temperatures. This temperature dependence of vapor pressure is vital for various applications, including the distillation process, predicting weather patterns, and even in the culinary field for understanding flavor infusion during cooking.