The temperature in kelvin at which the average speed of H2 molecues will be same as that of N2 molecules at 35 C will be ?

Answer

To solve the problem, we need to find the temperature in Kelvin at which the average speed of hydrogen (H₂) molecules will be the same as that of nitrogen (N₂) molecules at 35°C. We will use the formula for the average speed of gas molecules.

### Step-by-Step Solution:

1. **Convert the Given Temperature to Kelvin:**
- The temperature of nitrogen (Tn) is given as 35°C.
- To convert Celsius to Kelvin, we use the formula:
\[ T(K) = T(°C) + 273 \]
- Therefore,
\[ Tn = 35 + 273 = 308 \text{ K} \]

2. **Identify the Masses of the Molecules:**
- The mass of a hydrogen molecule (H₂) is approximately 2 amu (atomic mass units).
- The mass of a nitrogen molecule (N₂) is approximately 28 amu.

3. **Use the Formula for Average Speed:**
- The average speed (V) of gas molecules is given by:
\[ V = \sqrt{\frac{8RT}{\pi m}} \]
- Where:
- \( R \) is the gas constant,
- \( T \) is the temperature in Kelvin,
- \( m \) is the mass of the molecule.

4. **Set Up the Equation for Average Speeds:**
- According to the problem, the average speed of hydrogen (VH) equals the average speed of nitrogen (VN):
\[ V_H = V_N \]
- Substituting the formula for average speed:
\[ \sqrt{\frac{8R T_H}{\pi m_H}} = \sqrt{\frac{8R T_N}{\pi m_N}} \]

5. **Cancel Out Common Terms:**
- Since \( \sqrt{\frac{8R}{\pi}} \) is common on both sides, we can simplify the equation:
\[ \frac{T_H}{m_H} = \frac{T_N}{m_N} \]

6. **Rearranging to Find TH:**
- Rearranging gives us:
\[ T_H = T_N \cdot \frac{m_H}{m_N} \]

7. **Substituting the Known Values:**
- We know:
- \( T_N = 308 \text{ K} \)
- \( m_H = 2 \text{ amu} \)
- \( m_N = 28 \text{ amu} \)
- Substituting these values into the equation:
\[ T_H = 308 \cdot \frac{2}{28} \]

8. **Calculating TH:**
- Performing the calculation:
\[ T_H = 308 \cdot \frac{2}{28} = 308 \cdot \frac{1}{14} = 22 \text{ K} \]

### Final Answer:
The temperature in Kelvin at which the average speed of H₂ molecules will be the same as that of N₂ molecules at 35°C is **22 K**.