The average kinetic energy of a hydrogen molecule at `27^(@)C` is `9.3 xx 10^(-21) J`. The mass of hydrogen molecule is `3.1 xx 10^(-27) kg`. (i) Determine the average kinetic en-ergy at `227^(@)C`. (ii) Determine the root mean square speed of hydrogen molecule at `27^(@)C`.

Answer

To solve the problem, we will break it down into two parts as per the question:

### Part (i): Determine the average kinetic energy at `227^(@)C`.

1. **Convert the temperatures from Celsius to Kelvin:**
- The formula to convert Celsius to Kelvin is:
\[ T(K) = T(°C) + 273.15 \]
- For `27°C`:
\[ T_1 = 27 + 273.15 = 300.15 \, K \]
- For `227°C`:
\[ T_2 = 227 + 273.15 = 500.15 \, K \]

2. **Use the formula for average kinetic energy:**
The average kinetic energy (KE) of a molecule is given by:
\[ KE = \frac{3}{2} k T \]
where \( k \) is the Boltzmann constant, approximately \( 1.38 \times 10^{-23} \, J/K \).

3. **Calculate the average kinetic energy at `227°C`:**
\[ KE_2 = \frac{3}{2} k T_2 \]
Substituting the values:
\[ KE_2 = \frac{3}{2} \times (1.38 \times 10^{-23}) \times (500.15) \]
\[ KE_2 = \frac{3}{2} \times (1.38 \times 10^{-23}) \times (500.15) \approx 1.03 \times 10^{-20} \, J \]

### Part (ii): Determine the root mean square speed of hydrogen molecule at `27^(@)C`.

1. **Use the formula for root mean square speed (v_rms):**
The root mean square speed is given by:
\[ v_{rms} = \sqrt{\frac{3kT}{m}} \]
where \( m \) is the mass of the hydrogen molecule.

2. **Substitute the values:**
\[ v_{rms} = \sqrt{\frac{3 \times (1.38 \times 10^{-23}) \times (300.15)}{3.1 \times 10^{-27}}} \]

3. **Calculate the value:**
\[ v_{rms} = \sqrt{\frac{3 \times 1.38 \times 10^{-23} \times 300.15}{3.1 \times 10^{-27}}} \]
\[ v_{rms} \approx \sqrt{\frac{1.24 \times 10^{-20}}{3.1 \times 10^{-27}}} \approx \sqrt{4.0 \times 10^{6}} \approx 2000 \, m/s \]

### Final Answers:
(i) The average kinetic energy at `227°C` is approximately \( 1.03 \times 10^{-20} \, J \).

(ii) The root mean square speed of the hydrogen molecule at `27°C` is approximately \( 2000 \, m/s \).