The RMS velocity of `CH_(4)` will become double its value at STP when the temperature is :

Options

A. 546 K
B. `819^(@)C`
C. `546^(@)C`
D. 819 K

Explanation

To find the temperature at which the RMS velocity of \( CH_4 \) (methane) becomes double its value at Standard Temperature and Pressure (STP), we can follow these steps:

### Step 1: Understand the RMS Velocity Formula
The formula for the root mean square (RMS) velocity (\( v_{rms} \)) of a gas is given by:
\[ v_{rms} = \sqrt{\frac{3RT}{M}} \]
where:
- \( R \) is the universal gas constant,
- \( T \) is the temperature in Kelvin,
- \( M \) is the molar mass of the gas.

### Step 2: Identify the Molar Mass of \( CH_4 \)
The molar mass of methane (\( CH_4 \)) can be calculated as follows:
- Carbon (C) has a molar mass of approximately 12 g/mol.
- Hydrogen (H) has a molar mass of approximately 1 g/mol, and there are 4 hydrogen atoms in methane.

Thus, the molar mass \( M \) of \( CH_4 \) is:
\[ M = 12 + (4 \times 1) = 16 \, \text{g/mol} \]

### Step 3: Determine the RMS Velocity at STP
At STP, the temperature \( T \) is 273 K. Plugging this value into the RMS velocity formula gives:
\[ v_{rms, STP} = \sqrt{\frac{3R \cdot 273}{16}} \]

### Step 4: Set Up the Equation for Double the RMS Velocity
We want to find the temperature \( T' \) at which the RMS velocity is double that at STP:
\[ v_{rms, T'} = 2 \cdot v_{rms, STP} \]
Substituting the expression for \( v_{rms} \):
\[ \sqrt{\frac{3R T'}{16}} = 2 \cdot \sqrt{\frac{3R \cdot 273}{16}} \]

### Step 5: Simplify the Equation
Squaring both sides to eliminate the square root gives:
\[ \frac{3R T'}{16} = 4 \cdot \frac{3R \cdot 273}{16} \]
The \( \frac{3R}{16} \) cancels out:
\[ T' = 4 \cdot 273 \]

### Step 6: Calculate the New Temperature
Calculating \( T' \):
\[ T' = 4 \cdot 273 = 1092 \, \text{K} \]

### Step 7: Convert to Celsius
To convert from Kelvin to Celsius:
\[ T'_{Celsius} = 1092 - 273 = 819 \, \text{°C} \]

### Final Answer
The temperature at which the RMS velocity of \( CH_4 \) becomes double its value at STP is:
- **1092 K** or **819 °C**.

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