There is water in a closed container and the container is connected to a vacuum
pump. What would the temperature of the water in the container need to be to
cause the water to boil if the vacuum pump produced a pressure in the container
equal to 500% of the vapor pressure of water? What must the atmospheric pressure
be to cause water to boil at 212 degrees Fahrenheit?

To solve this problem, we need to understand two key concepts: the relationship between boiling point and pressure, and the vapor pressure of water.

### 1. Boiling Point and Pressure
The boiling point of water depends on the surrounding pressure. Water boils when its vapor pressure equals the surrounding atmospheric pressure.

### 2. Relationship Between Pressure and Boiling Point
Given that the pressure in the container is 500% (or 5 times) the vapor pressure of water, the boiling point can be determined from the known relationship between vapor pressure and temperature.

### Problem 1: Temperature for Boiling in the Container

Given:
- Pressure in the container = 500% of vapor pressure = \(5 \times P_{\text{vapor}}\)

We want to find the temperature \(T_{\text{boil}}\) at which water boils at this pressure. The vapor pressure of water at different temperatures is tabulated, but for simplicity, we can use the Clausius-Clapeyron equation, which approximates the relationship between temperature and vapor pressure:

\[
\ln\left(\frac{P_2}{P_1}\right) = -\frac{\Delta H_{\text{vap}}}{R} \left(\frac{1}{T_2} - \frac{1}{T_1}\right)
\]

Where:
- \(P_2 = 5 \times P_{\text{vapor}}\) at the unknown temperature \(T_2\)
- \(P_1 = P_{\text{vapor}}\) at a known temperature \(T_1\)
- \(\Delta H_{\text{vap}}\) = Heat of vaporization for water (approximately \(40.7 \, \text{kJ/mol}\))
- \(R\) = Gas constant \(8.314 \, \text{J/mol·K}\)

We can rearrange and solve for \(T_2\).

### Problem 2: Atmospheric Pressure for Boiling at 212°F (100°C)

Given:
- Boiling point = 212°F = 100°C
- Standard atmospheric pressure is \(1 \, \text{atm}\)

We want to find the pressure \(P_{\text{atmospheric}}\) required to cause water to boil at this temperature.

Using the vapor pressure of water at 100°C, which is 1 atm, we can directly infer that to cause boiling at this temperature, the atmospheric pressure must be 1 atm (101.3 kPa).

### Final Answers:
1. The temperature of the water in the container needed to cause it to boil at 5 times the vapor pressure can be found using the Clausius-Clapeyron equation. However, it is difficult to solve without numerical values or logarithmic tables. Typically, it requires a higher temperature than the normal boiling point.
  
2. The atmospheric pressure needed to cause water to boil at 212°F (100°C) is 1 atm.

Would you like a detailed calculation for the first part? Here are some related questions to consider:

1. What is the vapor pressure of water at different temperatures?
2. How does the Clausius-Clapeyron equation describe phase changes?
3. How does atmospheric pressure affect boiling points?
4. What is the significance of the heat of vaporization in boiling?
5. How would altitude affect the boiling point of water?

**Tip:** Understanding the relationship between temperature, pressure, and phase changes is crucial for thermodynamics and physical chemistry applications.

Explore how to determine the temperature required for water to boil under vacuum conditions using the Clausius-Clapeyron equation. Understand the relationship between pressure, temperature, and boiling points in thermodynamics.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation