On increasing the temperature, the root mean square speed of molecules of a gas filled in a container becomes double, now the pressure of the gas relative to the initial pressure will be

Options

A. equal
B. half
C. double
D. fout times

Explanation

To solve the problem, we need to analyze the relationship between the root mean square speed (VRMS) of gas molecules, temperature (T), and pressure (P) using the principles of the kinetic theory of gases.

### Step-by-Step Solution:

1. **Understanding the Relationship**:
The root mean square speed (VRMS) of gas molecules is given by the formula:
\[ V_{rms} = \sqrt{\frac{3RT}{M}} \]
where \(R\) is the universal gas constant, \(T\) is the absolute temperature, and \(M\) is the molar mass of the gas.

2. **Initial Conditions**:
Let the initial temperature be \(T_1\) and the initial root mean square speed be \(V_{rms1}\). According to the problem, when the temperature is increased, the root mean square speed becomes double:
\[ V_{rms2} = 2V_{rms1} \]

3. **Relating Temperature and VRMS**:
From the formula for VRMS, we can express the initial and final conditions:
\[ V_{rms1} = \sqrt{\frac{3RT_1}{M}} \quad \text{and} \quad V_{rms2} = \sqrt{\frac{3RT_2}{M}} \]
Setting \(V_{rms2} = 2V_{rms1}\):
\[ \sqrt{\frac{3RT_2}{M}} = 2 \sqrt{\frac{3RT_1}{M}} \]

4. **Squaring Both Sides**:
Squaring both sides gives:
\[ \frac{3RT_2}{M} = 4 \cdot \frac{3RT_1}{M} \]
Simplifying this, we find:
\[ T_2 = 4T_1 \]

5. **Using the Ideal Gas Law**:
The ideal gas law states:
\[ PV = nRT \]
For 1 mole of gas, this simplifies to:
\[ P_1V = RT_1 \quad \text{and} \quad P_2V = RT_2 \]
Dividing these two equations gives:
\[ \frac{P_2}{P_1} = \frac{T_2}{T_1} \]

6. **Substituting the Temperature Values**:
Since we found \(T_2 = 4T_1\):
\[ \frac{P_2}{P_1} = \frac{4T_1}{T_1} = 4 \]

7. **Conclusion**:
Therefore, the pressure of the gas relative to the initial pressure is:
\[ P_2 = 4P_1 \]

### Final Answer:
The pressure of the gas relative to the initial pressure will be **4 times the initial pressure**.

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