At what temperature the molecule of nitrogen will have same rms velocity as the molecule of oxygen at `127^(@)C` ?

Options

A. `457^(@C)`
B. `273^(@C)`
C. `350^(@C)`
D. `77^(@C)`

Answer

To solve the problem of finding the temperature at which nitrogen molecules will have the same RMS velocity as oxygen molecules at \(127^\circ C\), we can follow these steps:

### Step 1: Convert the temperature of oxygen from Celsius to Kelvin.
The formula to convert Celsius to Kelvin is:
\[ T(K) = T(°C) + 273 \]
For oxygen, the temperature is:
\[ T_{O_2} = 127 + 273 = 400 \, K \]

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

### Step 3: Set the RMS velocities of nitrogen and oxygen equal to each other.
Since we want the RMS velocities of nitrogen (\(V_{RMS,N_2}\)) and oxygen (\(V_{RMS,O_2}\)) to be equal, we can write:
\[ V_{RMS,N_2} = V_{RMS,O_2} \]
This gives us:
\[ \sqrt{\frac{3RT_{N_2}}{M_{N_2}}} = \sqrt{\frac{3RT_{O_2}}{M_{O_2}}} \]

### Step 4: Simplify the equation.
Squaring both sides and canceling out common terms, we have:
\[ \frac{T_{N_2}}{M_{N_2}} = \frac{T_{O_2}}{M_{O_2}} \]

### Step 5: Rearrange the equation to solve for \(T_{N_2}\).
Rearranging gives us:
\[ T_{N_2} = T_{O_2} \cdot \frac{M_{N_2}}{M_{O_2}} \]

### Step 6: Substitute known values.
We know:
- \(T_{O_2} = 400 \, K\)
- Molar mass of nitrogen (\(M_{N_2}\)) = 28 g/mol
- Molar mass of oxygen (\(M_{O_2}\)) = 32 g/mol

Substituting these values into the equation:
\[ T_{N_2} = 400 \cdot \frac{28}{32} \]

### Step 7: Calculate \(T_{N_2}\).
Calculating the right-hand side:
\[ T_{N_2} = 400 \cdot 0.875 = 350 \, K \]

### Step 8: Convert \(T_{N_2}\) from Kelvin to Celsius.
To convert Kelvin back to Celsius, we use:
\[ T(°C) = T(K) - 273 \]
Thus:
\[ T_{N_2} = 350 - 273 = 77 \, °C \]

### Final Answer:
The temperature at which nitrogen molecules will have the same RMS velocity as oxygen molecules at \(127^\circ C\) is \(77^\circ C\).

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