What is the temperature at which oxygen molecules have the same r.m.s. velocity as the hydrogen molecules at `27^(@)C`-

Options

A. `19.3 xx 10^4`
B. `17.8 xx 10^4`
C. `24.93 xx 10^9`
D. `17.8 xx 10^8`

Answer

To find the temperature at which oxygen molecules have the same root mean square (r.m.s.) velocity as hydrogen molecules at 27°C, we can follow these steps:

### Step 1: Understand the r.m.s. velocity formula
The r.m.s. velocity (u_rms) of a gas is given by the formula:
\[ u_{rms} = \sqrt{\frac{3RT}{M}} \]
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: Convert the temperature of hydrogen from Celsius to Kelvin
The temperature of hydrogen is given as 27°C. To convert this to Kelvin:
\[ T_{H_2} = 27 + 273 = 300 \text{ K} \]

### Step 3: Write the r.m.s. velocity equations for both gases
For hydrogen (H₂):
\[ u_{rms, H_2} = \sqrt{\frac{3RT_{H_2}}{M_{H_2}}} \]
For oxygen (O₂):
\[ u_{rms, O_2} = \sqrt{\frac{3RT_{O_2}}{M_{O_2}}} \]

### Step 4: Set the r.m.s. velocities equal to each other
Since we want the r.m.s. velocities to be equal:
\[ \sqrt{\frac{3RT_{H_2}}{M_{H_2}}} = \sqrt{\frac{3RT_{O_2}}{M_{O_2}}} \]

### Step 5: Cancel common terms
We can cancel \( \sqrt{3R} \) from both sides:
\[ \frac{T_{H_2}}{M_{H_2}} = \frac{T_{O_2}}{M_{O_2}} \]

### Step 6: Substitute the known values
The molar mass of hydrogen \( M_{H_2} = 2 \text{ g/mol} = 0.002 \text{ kg/mol} \) and the molar mass of oxygen \( M_{O_2} = 32 \text{ g/mol} = 0.032 \text{ kg/mol} \). Substituting these values:
\[ \frac{300}{0.002} = \frac{T_{O_2}}{0.032} \]

### Step 7: Solve for \( T_{O_2} \)
Cross-multiplying gives:
\[ 300 \times 0.032 = T_{O_2} \times 0.002 \]
\[ 9.6 = 0.002 T_{O_2} \]
\[ T_{O_2} = \frac{9.6}{0.002} = 4800 \text{ K} \]

### Step 8: Convert \( T_{O_2} \) back to Celsius
To convert Kelvin back to Celsius:
\[ T_{O_2} = 4800 - 273 = 4527 \text{ °C} \]

### Final Answer
The temperature at which oxygen molecules have the same r.m.s. velocity as hydrogen molecules at 27°C is **4527 °C**.

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