The remperature at which the `rms` speed of hydrogen molecules is equal to escape velocity on earth surface, will be

Options

A. `1060 K`
B. `5030 K`
C. `8270 K`
D. `10063 K`

Explanation

To find the temperature at which the root mean square (rms) speed of hydrogen molecules is equal to the escape velocity on the surface of the Earth, we can follow these steps:

### Step 1: Understand the relationship between rms speed and escape velocity
The root mean square speed (Vrms) of gas molecules is given by the formula:
\[ V_{\text{rms}} = \sqrt{\frac{3RT}{M}} \]
where:
- \( R \) is the universal gas constant (8.31 J/(mol·K)),
- \( T \) is the temperature in Kelvin,
- \( M \) is the molar mass of the gas in kg/mol.

The escape velocity (Ve) from the surface of the Earth is approximately 11.2 km/s or \( 11.2 \times 10^3 \) m/s.

### Step 2: Set up the equation
We set the rms speed equal to the escape velocity:
\[ \sqrt{\frac{3RT}{M}} = V_e \]

### Step 3: Rearrange the equation to solve for temperature (T)
Squaring both sides gives:
\[ \frac{3RT}{M} = V_e^2 \]
Now, rearranging for \( T \):
\[ T = \frac{M V_e^2}{3R} \]

### Step 4: Substitute the values
For hydrogen, the molar mass \( M \) is 2 g/mol, which is \( 2 \times 10^{-3} \) kg/mol. The escape velocity \( V_e \) is \( 11.2 \times 10^3 \) m/s. Now we can substitute these values into the equation:
\[ T = \frac{(2 \times 10^{-3} \, \text{kg/mol}) \times (11.2 \times 10^3 \, \text{m/s})^2}{3 \times 8.31 \, \text{J/(mol·K)}} \]

### Step 5: Calculate the temperature
Calculating \( V_e^2 \):
\[ V_e^2 = (11.2 \times 10^3)^2 = 1.2544 \times 10^8 \, \text{m}^2/\text{s}^2 \]
Now substituting this back into the equation for \( T \):
\[ T = \frac{(2 \times 10^{-3}) \times (1.2544 \times 10^8)}{3 \times 8.31} \]
Calculating the numerator:
\[ 2 \times 10^{-3} \times 1.2544 \times 10^8 = 250.88 \]
Now calculating the denominator:
\[ 3 \times 8.31 = 24.93 \]
Finally, calculating \( T \):
\[ T = \frac{250.88}{24.93} \approx 10.07 \times 10^3 \approx 10075 \, \text{K} \]

### Final Answer
The temperature at which the rms speed of hydrogen molecules is equal to the escape velocity on the surface of the Earth is approximately **10075 K**.