Find the temeprature at which the r.m.s. velocity is equal to the escape velocity from the surface of the earth for hydrogen. Escape velocity `= 11.2 km//s `

Explanation

To find the temperature at which the root mean square (r.m.s.) velocity of hydrogen is equal to the escape velocity from the surface of the Earth, we can follow these steps:

### Step 1: Understand the Given Values
- Escape velocity (V_escape) = 11.2 km/s
- Convert this to meters per second:
\[ V_{\text{escape}} = 11.2 \, \text{km/s} = 11.2 \times 10^3 \, \text{m/s} \]

### Step 2: Use the Formula for r.m.s. Velocity
The r.m.s. velocity (V_rms) for an ideal gas is given by the formula:
\[ V_{\text{rms}} = \sqrt{\frac{3RT}{M}} \]
Where:
- \( R \) = gas constant = 8.31 J/(mol·K)
- \( T \) = temperature in Kelvin
- \( M \) = molar mass of the gas in kg/mol

### Step 3: Identify the Molar Mass of Hydrogen
The molar mass of hydrogen (H₂) is approximately 2 g/mol. Convert this to kg:
\[ M = 2 \, \text{g/mol} = 2 \times 10^{-3} \, \text{kg/mol} \]

### Step 4: Set r.m.s. Velocity Equal to Escape Velocity
Since we want the r.m.s. velocity to equal the escape velocity, we set:
\[ V_{\text{rms}} = V_{\text{escape}} = 11.2 \times 10^3 \, \text{m/s} \]

### Step 5: Square Both Sides of the Equation
Squaring both sides gives:
\[ V_{\text{rms}}^2 = \frac{3RT}{M} \]
Substituting \( V_{\text{escape}} \):
\[ (11.2 \times 10^3)^2 = \frac{3RT}{M} \]

### Step 6: Rearrange to Solve for Temperature (T)
Rearranging the equation for \( T \):
\[ T = \frac{V_{\text{rms}}^2 \cdot M}{3R} \]

### Step 7: Substitute the Values
Substituting the known values into the equation:
\[ T = \frac{(11.2 \times 10^3)^2 \cdot (2 \times 10^{-3})}{3 \cdot 8.31} \]

### Step 8: Calculate the Value
Calculating \( (11.2 \times 10^3)^2 \):
\[ (11.2 \times 10^3)^2 = 125.44 \times 10^6 = 1.2544 \times 10^8 \, \text{m}^2/\text{s}^2 \]
Now substituting back:
\[ T = \frac{(1.2544 \times 10^8) \cdot (2 \times 10^{-3})}{3 \cdot 8.31} \]
Calculating the denominator:
\[ 3 \cdot 8.31 = 24.93 \]
Now substituting:
\[ T = \frac{(1.2544 \times 10^8) \cdot (2 \times 10^{-3})}{24.93} \]
Calculating the numerator:
\[ 1.2544 \times 10^8 \cdot 2 \times 10^{-3} = 2.5088 \times 10^5 \]
Finally, calculating \( T \):
\[ T = \frac{2.5088 \times 10^5}{24.93} \approx 10063 \, \text{K} \]

### Final Answer
The temperature at which the r.m.s. velocity of hydrogen is equal to the escape velocity from the surface of the Earth is approximately **10063 K**.

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