The `rms` velocity of `CO_(2)` at temperature T(in Kelvin) is x cm `s^(-1)`. At what temperature (in Kelvin) would the `rms` velocity of nitrous oxide be `4 x cm s^(-1)`?

Options

A. 16T
B. 2T
C. 4T
D. 32T

Answer

To solve the problem, we need to find the temperature at which the root mean square (rms) velocity of nitrous oxide (N2O) is 4 times that of carbon dioxide (CO2). Let's break down the solution step by step.

### Step-by-Step Solution:

1. **Understanding the RMS Velocity Formula**:
The rms velocity (u_rms) of a gas is given by the formula:
\[ u_{\text{rms}} = \sqrt{\frac{3RT}{M}} \]
where:
- \( R \) is the universal gas constant,
- \( T \) is the absolute temperature in Kelvin,
- \( M \) is the molar mass of the gas.

2. **Setting Up the Equation**:
For CO2 at temperature \( T \), the rms velocity is given as \( x \) cm/s:
\[ u_{\text{rms, CO2}} = x = \sqrt{\frac{3RT}{M_{\text{CO2}}}} \]
For N2O, we want the rms velocity to be \( 4x \) cm/s:
\[ u_{\text{rms, N2O}} = 4x = \sqrt{\frac{3RT_{\text{N2O}}}{M_{\text{N2O}}}} \]

3. **Finding Molar Masses**:
- Molar mass of CO2 (\( M_{\text{CO2}} \)):
\[ M_{\text{CO2}} = 12 + 2 \times 16 = 44 \text{ g/mol} \]
- Molar mass of N2O (\( M_{\text{N2O}} \)):
\[ M_{\text{N2O}} = 14 \times 2 + 16 = 28 + 16 = 44 \text{ g/mol} \]

4. **Setting Up the Ratio of RMS Velocities**:
We can set up the ratio of the rms velocities:
\[ \frac{u_{\text{rms, CO2}}}{u_{\text{rms, N2O}}} = \sqrt{\frac{T_{\text{CO2}} \cdot M_{\text{N2O}}}{T_{\text{N2O}} \cdot M_{\text{CO2}}}} \]
Substituting the known values:
\[ \frac{x}{4x} = \sqrt{\frac{T \cdot 44}{T_{\text{N2O}} \cdot 44}} \]
This simplifies to:
\[ \frac{1}{4} = \sqrt{\frac{T}{T_{\text{N2O}}}} \]

5. **Squaring Both Sides**:
Squaring both sides gives:
\[ \left(\frac{1}{4}\right)^2 = \frac{T}{T_{\text{N2O}}} \]
\[ \frac{1}{16} = \frac{T}{T_{\text{N2O}}} \]

6. **Finding Temperature of N2O**:
Rearranging gives:
\[ T_{\text{N2O}} = 16T \]

### Final Answer:
The temperature at which the rms velocity of nitrous oxide (N2O) would be \( 4x \) cm/s is:
\[ T_{\text{N2O}} = 16T \text{ K} \]