https://file.aiquickdraw.com/mathbot/file/1730689254600xucx640b.jpg

To solve this problem, we'll use the **Clausius-Clapeyron equation** to estimate the vapor pressure of pentane at a given temperature. Here's the information provided:

- Normal boiling point (\( T_1 \)) of pentane: 36.0°C = 309.15 K
- Vapor pressure at the normal boiling point (\( P_1 \)): 1 atm (since this is the boiling point, where the vapor pressure equals atmospheric pressure)
- Temperature of interest (\( T_2 \)): 5.0°C = 278.15 K
- Enthalpy of vaporization (\( \Delta H_{vap} \)): 27.6 kJ/mol
- Desired pressure units: Torr

### Step-by-Step Solution:

1. **Convert Enthalpy to J/mol**: Since the enthalpy of vaporization (\( \Delta H_{vap} \)) is given in kJ/mol, convert it to J/mol for use in the Clausius-Clapeyron equation.
   \[
   \Delta H_{vap} = 27.6 \, \text{kJ/mol} = 27600 \, \text{J/mol}
   \]

2. **Use the Clausius-Clapeyron Equation**:
   \[
   \ln \left( \frac{P_2}{P_1} \right) = -\frac{\Delta H_{vap}}{R} \left( \frac{1}{T_2} - \frac{1}{T_1} \right)
   \]
   where:
   - \( R = 8.314 \, \text{J/(mol·K)} \) (ideal gas constant)
   - \( P_1 = 1 \, \text{atm} \)
   - \( T_1 = 309.15 \, \text{K} \)
   - \( T_2 = 278.15 \, \text{K} \)

3. **Calculate \( \ln \left( \frac{P_2}{P_1} \right) \)**:
   Substitute the values:
   \[
   \ln \left( \frac{P_2}{1 \, \text{atm}} \right) = -\frac{27600}{8.314} \left( \frac{1}{278.15} - \frac{1}{309.15} \right)
   \]

4. **Solve for \( P_2 \)**:
   - Compute the right side of the equation to find the value of \( \ln \left( \frac{P_2}{P_1} \right) \).
   - Then, exponentiate both sides to solve for \( P_2 \).

5. **Convert \( P_2 \) to Torr**: Since \( 1 \, \text{atm} = 760 \, \text{Torr} \), multiply the result by 760 to get the pressure in Torr.

Let me calculate these steps for you.The vapor pressure of pentane at 5.0°C is approximately **229.65 Torr**.

Would you like further details on any of the steps, or do you have any questions?

Here are some related questions you might consider:
1. How does the Clausius-Clapeyron equation work for estimating vapor pressures at different temperatures?
2. What are some practical applications of knowing vapor pressures for substances like pentane?
3. How does the enthalpy of vaporization affect the vapor pressure?
4. What units are typically used for vapor pressure, and why might Torr be preferred in certain cases?
5. How can the Clausius-Clapeyron equation be used to find the boiling point of a substance under different pressures?

**Tip:** When using the Clausius-Clapeyron equation, always ensure temperature is in Kelvin and energy in consistent units (e.g., J/mol with R in J/(mol·K)).

Determine the vapor pressure of pentane on a morning when the temperature is 5.0°C. Units requested for the pressure are in Torr.

Learn how to calculate the vapor pressure of pentane at 5.0°C using the Clausius-Clapeyron equation. This problem involves key thermodynamic concepts, including enthalpy of vaporization and phase equilibrium, making it suitable for high school chemistry students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation