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. reduce 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 states:

\[ PV = nRT \]

Where:
- \( P \) = pressure
- \( V \) = volume
- \( n \) = number of moles of gas (constant for this problem)
- \( R \) = ideal gas constant (constant for this problem)
- \( T \) = temperature in Kelvin

### Step-by-Step Solution:

1. **Understand the Relationship**:
From the ideal gas law, we can derive that the volume \( V \) is directly proportional to the temperature \( T \) and inversely proportional to the pressure \( P \):
\[ V \propto \frac{T}{P} \]

2. **Set Up the Equation**:
To find the conditions under which the volume becomes four times its original value, we can express this mathematically:
\[ V_2 = 4V_1 \]
This can be rewritten using the proportionality:
\[ \frac{V_2}{V_1} = \frac{T_2}{T_1} \cdot \frac{P_1}{P_2} \]

3. **Substituting Values**:
We need to manipulate the equation to achieve \( \frac{V_2}{V_1} = 4 \). This means:
\[ 4 = \frac{T_2}{T_1} \cdot \frac{P_1}{P_2} \]

4. **Choose a Method**:
Let's consider the first option: **Double the temperature and reduce the pressure to half**.
- If we double the temperature:
\[ T_2 = 2T_1 \]
- If we reduce the pressure to half:
\[ P_2 = \frac{1}{2}P_1 \]

5. **Calculate the New Volume**:
Substitute these values into the equation:
\[ \frac{V_2}{V_1} = \frac{2T_1}{T_1} \cdot \frac{P_1}{\frac{1}{2}P_1} = 2 \cdot 2 = 4 \]
Thus, \( V_2 = 4V_1 \).

6. **Evaluate Other Options**:
- **Option 2**: Double the temperature and double the pressure. This gives:
\[ \frac{V_2}{V_1} = \frac{2T_1}{T_1} \cdot \frac{P_1}{2P_1} = 2 \cdot \frac{1}{2} = 1 \quad (\text{Incorrect}) \]
- **Option 3**: Reduce the temperature to half and double the pressure. This gives:
\[ \frac{V_2}{V_1} = \frac{\frac{1}{2}T_1}{T_1} \cdot \frac{P_1}{2P_1} = \frac{1}{2} \cdot \frac{1}{2} = \frac{1}{4} \quad (\text{Incorrect}) \]
- **Option 4**: Reduce the temperature to half and reduce the pressure to half. This gives:
\[ \frac{V_2}{V_1} = \frac{\frac{1}{2}T_1}{T_1} \cdot \frac{P_1}{\frac{1}{2}P_1} = \frac{1}{2} \cdot 2 = 1 \quad (\text{Incorrect}) \]

### Conclusion:
The only correct option that allows the volume of the gas to become four times its original volume is to **double the temperature and reduce the pressure to half**.