The temperature of `H_(2)` gas at which `rms` speed of `H_(2)` molecules is equal to escape speed of a particle from earth surface is :

Options

A. `10^(3)K`
B. `2xx10^(3)K`
C. `10^(4)K`
D. `10^(5)K`

Answer

To find the temperature of \( H_2 \) gas at which the root mean square (rms) speed of \( H_2 \) molecules is equal to the escape speed from the Earth's surface, we can follow these steps:

### Step 1: Understand the escape speed
The escape speed from the Earth's surface is given as \( 11.2 \, \text{km/s} \). We convert this to meters per second:
\[ v_e = 11.2 \times 10^3 \, \text{m/s} \]

### Step 2: Write the formula for rms speed
The rms speed (\( v_{rms} \)) of gas molecules is given by the formula:
\[ v_{rms} = \sqrt{\frac{3RT}{M}} \]
where:
- \( R \) is the universal gas constant (\( R = 8.314 \, \text{J/(mol K)} \)),
- \( T \) is the temperature in Kelvin,
- \( M \) is the molar mass of the gas in kg/mol.

### Step 3: Determine the molar mass of \( H_2 \)
The molar mass of \( H_2 \) is approximately \( 2 \, \text{g/mol} \), which we convert to kg:
\[ M = 2 \times 10^{-3} \, \text{kg/mol} \]

### Step 4: Set the rms speed equal to escape speed
We set the rms speed equal to the escape speed:
\[ \sqrt{\frac{3RT}{M}} = 11.2 \times 10^3 \]

### Step 5: Square both sides
Squaring both sides to eliminate the square root gives:
\[ \frac{3RT}{M} = (11.2 \times 10^3)^2 \]

### Step 6: Solve for temperature \( T \)
Rearranging the equation to solve for \( T \):
\[ T = \frac{M (11.2 \times 10^3)^2}{3R} \]

### Step 7: Substitute values
Substituting the values of \( M \) and \( R \):
\[ T = \frac{(2 \times 10^{-3}) (11.2 \times 10^3)^2}{3 \times 8.314} \]

### Step 8: Calculate \( T \)
Calculating the right-hand side:
1. Calculate \( (11.2 \times 10^3)^2 = 125.44 \times 10^6 = 1.2544 \times 10^8 \).
2. Substitute this value into the equation:
\[ T = \frac{(2 \times 10^{-3}) (1.2544 \times 10^8)}{3 \times 8.314} \]
3. Calculate \( 3 \times 8.314 = 24.942 \).
4. Finally, compute:
\[ T = \frac{(2.5088 \times 10^5)}{24.942} \approx 10035.2 \, \text{K} \]

### Conclusion
Thus, the temperature of \( H_2 \) gas at which the rms speed of \( H_2 \) molecules is equal to the escape speed from Earth is approximately \( 10035.2 \, \text{K} \).