Which of the following methods will enable the volume of an ideal gas to be made four times

Options

A. Double the temperature and reduce the pressure to half
B. Double the temperature and also double the pressure
C. Reduced the temperature to half and double the pressure
D. Reduce the temperature to half and reduce the pressure to half

Answer

To solve the problem of how to make the volume of an ideal gas four times its original volume, we can use the Ideal Gas Law, which is given by the equation:

\[ PV = nRT \]

where:
- \( P \) = pressure
- \( V \) = volume
- \( n \) = number of moles of gas
- \( R \) = universal gas constant
- \( T \) = temperature in Kelvin

### Step-by-step Solution:

1. **Understand the Initial Conditions**:
- Let the initial pressure be \( P_0 \), initial volume be \( V_0 \), and initial temperature be \( T_0 \).
- We want to find a method to make the final volume \( V_f = 4V_0 \).

2. **Option 1: Double the Temperature and Reduce Pressure to Half**:
- If we double the temperature: \( T_f = 2T_0 \)
- If we reduce the pressure to half: \( P_f = \frac{P_0}{2} \)
- Using the Ideal Gas Law:
\[ P_f V_f = nR T_f \]
Substituting the values:
\[ \frac{P_0}{2} V_f = nR (2T_0) \]
Rearranging gives:
\[ V_f = \frac{2nRT_0}{\frac{P_0}{2}} = \frac{4nRT_0}{P_0} \]
Since \( nRT_0 = P_0 V_0 \), we can substitute:
\[ V_f = \frac{4P_0 V_0}{P_0} = 4V_0 \]
- Therefore, this option is correct.

3. **Option 2: Double the Temperature and Double the Pressure**:
- If we double the temperature: \( T_f = 2T_0 \)
- If we double the pressure: \( P_f = 2P_0 \)
- Using the Ideal Gas Law:
\[ P_f V_f = nR T_f \]
Substituting the values:
\[ 2P_0 V_f = nR (2T_0) \]
Rearranging gives:
\[ V_f = \frac{nRT_0}{P_0} = V_0 \]
- Therefore, this option is incorrect.

4. **Option 3: Reduce the Temperature to Half and Double the Pressure**:
- If we reduce the temperature to half: \( T_f = \frac{T_0}{2} \)
- If we double the pressure: \( P_f = 2P_0 \)
- Using the Ideal Gas Law:
\[ P_f V_f = nR T_f \]
Substituting the values:
\[ 2P_0 V_f = nR \left(\frac{T_0}{2}\right) \]
Rearranging gives:
\[ V_f = \frac{nRT_0}{4P_0} = \frac{V_0}{4} \]
- Therefore, this option is incorrect.

5. **Option 4: Reduce the Temperature to Half and Reduce the Pressure to Half**:
- If we reduce the temperature to half: \( T_f = \frac{T_0}{2} \)
- If we reduce the pressure to half: \( P_f = \frac{P_0}{2} \)
- Using the Ideal Gas Law:
\[ P_f V_f = nR T_f \]
Substituting the values:
\[ \frac{P_0}{2} V_f = nR \left(\frac{T_0}{2}\right) \]
Rearranging gives:
\[ V_f = \frac{nRT_0}{P_0} = V_0 \]
- Therefore, this option is incorrect.

### Conclusion:
The only method that enables the volume of the ideal gas to be made four times is **Option 1: Double the temperature and reduce the pressure to half**.