At what temperature is the rms velocity of a hydrogen molecule equal to that of an oxygen molecule at `47^(@)C`?

Options

A. `80 K`
B. `-73 K`
C. `3 K `
D. `20 K`

Explanation

To find the temperature at which the root mean square (rms) velocity of a hydrogen molecule is equal to that of an oxygen molecule at \(47^\circ C\), we can follow these steps:

### Step 1: Convert the temperature of oxygen from Celsius to Kelvin
The temperature in Kelvin can be calculated using the formula:
\[ T(K) = T(°C) + 273 \]
For oxygen at \(47^\circ C\):
\[ T_{O_2} = 47 + 273 = 320 \, K \]

### Step 2: Write the formula for rms velocity
The rms velocity (\(v_{rms}\)) of a gas is given by the formula:
\[ v_{rms} = \sqrt{\frac{3RT}{M}} \]
where \(R\) is the universal gas constant, \(T\) is the temperature in Kelvin, and \(M\) is the molar mass of the gas.

### Step 3: Set the rms velocities of hydrogen and oxygen equal to each other
Let \(T_{H_2}\) be the temperature of hydrogen. The rms velocities of hydrogen and oxygen can be expressed as:
\[ v_{rms, H_2} = \sqrt{\frac{3RT_{H_2}}{M_{H_2}}} \]
\[ v_{rms, O_2} = \sqrt{\frac{3RT_{O_2}}{M_{O_2}}} \]
Setting these equal gives:
\[ \sqrt{\frac{3RT_{H_2}}{M_{H_2}}} = \sqrt{\frac{3RT_{O_2}}{M_{O_2}}} \]

### Step 4: Square both sides and simplify
Squaring both sides removes the square roots:
\[ \frac{3RT_{H_2}}{M_{H_2}} = \frac{3RT_{O_2}}{M_{O_2}} \]
The \(3R\) cancels out:
\[ \frac{T_{H_2}}{M_{H_2}} = \frac{T_{O_2}}{M_{O_2}} \]

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

### Step 6: Substitute known values
The molar mass of hydrogen (\(H_2\)) is \(2 \, g/mol\) and the molar mass of oxygen (\(O_2\)) is \(32 \, g/mol\):
\[ T_{H_2} = 320 \cdot \frac{2}{32} \]

### Step 7: Calculate \(T_{H_2}\)
Calculating this gives:
\[ T_{H_2} = 320 \cdot \frac{2}{32} = 320 \cdot \frac{1}{16} = 20 \, K \]

Thus, the temperature at which the rms velocity of a hydrogen molecule is equal to that of an oxygen molecule at \(47^\circ C\) is \(20 \, K\).