Lecture Notes: Clausius-Clapeyron Equation

Overview

• Focus on the Clausius-Clapeyron equation.
• Explore different forms of the equation.
• Examples and practice problems for application.

Main Form of the Equation

• The equation: [ \ln\left(\frac{P2}{P1}\right) = -\frac{\Delta H_{vap}}{R} \left(\frac{1}{T2} - \frac{1}{T1}\right) ]
  • Where:
    • ( \Delta H_{vap} ) = Enthalpy of vaporization (in Joules per mole).
    • ( R ) = Energy constant (8.3145 J/mol K).
    • ( T ) = Temperature in Kelvin.

Key Concepts

• Unit Conversion:
  • Convert kilojoules to joules by multiplying by 1000.
  • Convert Celsius to Kelvin by adding 273.
• Equation Variations:
  • Different forms exist, sometimes with a positive or negative sign.
  • Flipping pressures or temperatures can change the sign.

Alternative Forms of the Equation

1. Finding a new vapor pressure (P2): [ P2 = P1 \times e^{\left(-\frac{\Delta H_{vap}}{R} \left(\frac{1}{T2} - \frac{1}{T1}\right)\right)} ]

   • E is the base of the natural logarithm (~2.718).

2. Calculating heat of vaporization: [ \Delta H_{vap} = -R \times \ln\left(\frac{P2}{P1}\right) \div \left(\frac{1}{T2} - \frac{1}{T1}\right) ]

3. Finding the second temperature (T2): [ T2 = \left( \frac{1}{T1} - \frac{R \ln\left(\frac{P2}{P1}\right)}{\Delta H_{vap}} \right)^{-1} ]

Example Problems

Problem 1

• Given:
  • ( P1 = 21 ) Torr at ( T1 = 300 ) K.
  • Find ( P2 ) at ( T2 = 310 ) K.
  • ( \Delta H_{vap} = 24 ) kJ/mol (convert to 24,000 J/mol).
• Solution Steps:
  1. Use the equation ( P2 = P1 \times e^{\left(-\frac{\Delta H_{vap}}{R} \left(\frac{1}{T2} - \frac{1}{T1}\right)\right)} ).
  2. Calculate each component step by step.
  3. Final ( P2 ) = 28.6 Torr.
• Conclusion: Vapor pressure increases with temperature.

Problem 2

• Given:
  • ( P1 = 30 ) Torr at ( T1 = 250 ) K.
  • ( P2 = 150 ) Torr.
  • ( \Delta H_{vap} = 45 ) kJ/mol (convert to 45,000 J/mol).
• Solution Steps:
  1. Use the equation for ( T2 ): [ T2 = \left( \frac{1}{T1} - \frac{R \ln\left(\frac{P2}{P1}\right)}{\Delta H_{vap}} \right)^{-1} ]
  2. Calculate step by step.
  3. Final ( T2 ) = 270.1 K.
• Conclusion: Higher vapor pressure correlates with higher temperature.

Summary

• Understanding different forms of the Clausius-Clapeyron equation is essential.
• Proper unit conversion and careful calculation steps are crucial for accuracy.
• Practice problems illustrate the relationship between temperature and vapor pressure.