The tempeature at which nitrogen under 1 atmospheric pressure has the same root mean square velocity as that of `CO_(2)` at STP is

Options

A. `-99.27^(@)C`
B. `-99.27 K`
C. `-9.927^(@)C`
D. `156^(@)C`

Explanation

To find the temperature at which nitrogen under 1 atmospheric pressure has the same root mean square (RMS) velocity as that of carbon dioxide (CO₂) at standard temperature and pressure (STP), we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Formula for RMS Velocity**:
The root mean square velocity (VRMS) 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 in kg/mol.

2. **Set Up the Equation**:
We need to equate the RMS velocities of nitrogen (N₂) and carbon dioxide (CO₂):
\[ V_{RMS, N_2} = V_{RMS, CO_2} \]
Thus, we have:
\[ \sqrt{\frac{3RT_{N_2}}{M_{N_2}}} = \sqrt{\frac{3RT_{CO_2}}{M_{CO_2}}} \]

3. **Cancel Out Common Terms**:
Since \( R \) and \( 3 \) are common on both sides, we can cancel them:
\[ \sqrt{\frac{T_{N_2}}{M_{N_2}}} = \sqrt{\frac{T_{CO_2}}{M_{CO_2}}} \]

4. **Square Both Sides**:
Squaring both sides to eliminate the square root gives:
\[ \frac{T_{N_2}}{M_{N_2}} = \frac{T_{CO_2}}{M_{CO_2}} \]

5. **Rearranging the Equation**:
Rearranging the equation to solve for \( T_{N_2} \):
\[ T_{N_2} = T_{CO_2} \times \frac{M_{N_2}}{M_{CO_2}} \]

6. **Substituting Known Values**:
- The molar mass of nitrogen, \( M_{N_2} = 28 \, \text{g/mol} \).
- The molar mass of carbon dioxide, \( M_{CO_2} = 44 \, \text{g/mol} \).
- The temperature of CO₂ at STP, \( T_{CO_2} = 273 \, \text{K} \).

Substituting these values into the equation:
\[ T_{N_2} = 273 \times \frac{28}{44} \]

7. **Calculating the Temperature**:
\[ T_{N_2} = 273 \times 0.63636 \approx 173.37 \, \text{K} \]

8. **Convert Kelvin to Celsius**:
To convert the temperature from Kelvin to Celsius:
\[ T_{C} = T_{N_2} - 273.15 \approx 173.37 - 273.15 \approx -99.78 \, \text{°C} \]

### Final Answer:
The temperature at which nitrogen under 1 atmospheric pressure has the same root mean square velocity as that of CO₂ at STP is approximately **-99.78 °C**.