Clausius Clayperon Equations

Clausius-Clapeyron equation describes the phase transition between two phases of a substance, typically the liquid and vapor phases. This equation provides a mathematical relationship between the vapor pressure of a substance and its temperature.

The equation is given as:

$\frac{d \ln P}{dT} = \frac{\Delta H_v}{R \cdot T^2}$

Where:
– $\frac{d \ln P}{dT}$ is the rate of change of the natural logarithm of the vapor pressure ( $P$ ) with respect to temperature ( $T$ ).
– $\Delta H_v$ is the enthalpy of vaporization (latent heat) of the substance, representing the energy required to change a unit quantity of the substance from liquid to vapor.
– $R$ is the universal gas constant.

Integrating this equation over a range of temperatures gives the more common form:

$\ln \left( \frac{P_2}{P_1} \right) = -\frac{\Delta H_v}{R} \left( \frac{1}{T_2} - \frac{1}{T_1} \right)$

Where $P_1$ and $P_2$ are the vapor pressures at temperatures $T_1$ and $T_2$ , respectively.

This equation is used to predict changes in boiling points under different pressures or to calculate the vapor pressure at a specific temperature given some reference data.

Problems

1. **Problem 1:**
The vapor pressure of water at $373 \, \text{K}$ (100°C) is $101.3 \, \text{kPa}$ . Given that the enthalpy of vaporization for water is $40.79 \, \text{kJ/mol}$ , calculate the vapor pressure of water at $363 \, \text{K}$ .

2. **Problem 2:**
A substance has a vapor pressure of $50 \, \text{kPa}$ at $300 \, \text{K}$ and $70 \, \text{kPa}$ at $310 \, \text{K}$ . Determine the enthalpy of vaporization for this substance.

3. **Problem 3:**
Using the information from problem 2, calculate the temperature at which the vapor pressure of the substance would be $100 \, \text{kPa}$ .

4. **Problem 4:**
The enthalpy of vaporization for ethanol is $38.56 \, \text{kJ/mol}$ . If the vapor pressure of ethanol is $78 \, \text{kPa}$ at $351 \, \text{K}$ , find the vapor pressure of ethanol at $360 \, \text{K}$ .

5. **Problem 5:**
Benzene has a boiling point of $353 \, \text{K}$ at atmospheric pressure ( $101.3 \, \text{kPa}$ ). If the enthalpy of vaporization is $30.8 \, \text{kJ/mol}$ , at what temperature will benzene boil if the external pressure is $50 \, \text{kPa}$ ?

6. **Problem 6:**
Given that the vapor pressure of a liquid increases by 15% when the temperature is increased by $5 \, \text{K}$ and its enthalpy of vaporization is $25 \, \text{kJ/mol}$ , determine the initial vapor pressure of the liquid.
Key

For each problem, you can utilize the Clausius-Clapeyron equation, and some may require manipulation or rearrangement of the equation to solve for the desired variable.

1. **Problem 1:** $P = 70.11 \, \text{kPa}$