At what temperature is the rms speed of `H_(2)` molecules the same as that of oxygen molecules at `1327^(@)C` ?

Answer

To solve the problem of finding the temperature at which the root mean square (rms) speed of \( H_2 \) molecules is the same as that of \( O_2 \) molecules at \( 1327^\circ C \), we can follow these steps:

### Step 1: Understand the formula for rms speed
The rms speed (\( v_{rms} \)) of a gas is given by the formula:
\[ v_{rms} = \sqrt{\frac{3RT}{M}} \]
where:
- \( R \) is the universal gas constant,
- \( T \) is the absolute temperature in Kelvin,
- \( M \) is the molar mass of the gas in kg/mol.

### Step 2: Set up the equation for both gases
We need to find the temperature \( T_{H_2} \) at which the rms speed of \( H_2 \) is equal to that of \( O_2 \). Thus, we can set up the equation:
\[ \sqrt{\frac{3RT_{H_2}}{M_{H_2}}} = \sqrt{\frac{3RT_{O_2}}{M_{O_2}}} \]

### Step 3: Square both sides to eliminate the square root
Squaring both sides gives:
\[ \frac{3RT_{H_2}}{M_{H_2}} = \frac{3RT_{O_2}}{M_{O_2}} \]

### Step 4: Cancel common terms
We can cancel \( 3R \) from both sides:
\[ \frac{T_{H_2}}{M_{H_2}} = \frac{T_{O_2}}{M_{O_2}} \]

### Step 5: Rearrange the equation to find \( T_{H_2} \)
Rearranging gives:
\[ T_{H_2} = T_{O_2} \cdot \frac{M_{H_2}}{M_{O_2}} \]

### Step 6: Identify the molar masses
The molar mass of \( H_2 \) is \( 2 \, \text{g/mol} \) and the molar mass of \( O_2 \) is \( 32 \, \text{g/mol} \).

### Step 7: Convert the temperature of \( O_2 \) to Kelvin
The given temperature of \( O_2 \) is \( 1327^\circ C \). To convert this to Kelvin:
\[ T_{O_2} = 1327 + 273 = 1600 \, K \]

### Step 8: Substitute the values into the equation
Substituting the values into our equation:
\[ T_{H_2} = 1600 \cdot \frac{2}{32} \]

### Step 9: Calculate \( T_{H_2} \)
Calculating the right-hand side:
\[ T_{H_2} = 1600 \cdot \frac{1}{16} = 100 \, K \]

### Final Answer
The temperature at which the rms speed of \( H_2 \) molecules is the same as that of \( O_2 \) molecules at \( 1327^\circ C \) is:
\[ \boxed{100 \, K} \]