If the rms speed of the nitrogen molecules of the gas at room temperature is 500 m/s, then the rms speed of the hydrogen molecules at the same temperature will be –

Options

A. 1870 m/s
B. 1935 m/s
C. 7000 m/s
D. 83.7 m/s

Answer

To solve the problem of finding the root mean square (RMS) speed of hydrogen molecules at room temperature, given the RMS speed of nitrogen molecules, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the RMS Speed Formula**:
The formula for the root mean square speed (VRMS) of a gas is given by:
\[ V_{RMS} = \sqrt{\frac{3RT}{M}} \]
where:
- \( R \) is the gas constant,
- \( T \) is the temperature in Kelvin,
- \( M \) is the molar mass of the gas in kg.

2. **Given Data for Nitrogen (N2)**:
- The RMS speed of nitrogen molecules, \( V_{RMS, N2} = 500 \, \text{m/s} \).
- The molar mass of nitrogen \( M_{N2} = 28 \, \text{g/mol} = 28 \times 10^{-3} \, \text{kg/mol} \).

3. **Setting Up the Equation for Nitrogen**:
Plugging the values into the RMS speed formula for nitrogen:
\[ 500 = \sqrt{\frac{3RT}{28 \times 10^{-3}}} \]
Squaring both sides:
\[ 500^2 = \frac{3RT}{28 \times 10^{-3}} \]
This can be rearranged to express \( 3RT \):
\[ 3RT = 500^2 \times 28 \times 10^{-3} \]

4. **Finding the RMS Speed for Hydrogen (H2)**:
- The molar mass of hydrogen \( M_{H2} = 2 \, \text{g/mol} = 2 \times 10^{-3} \, \text{kg/mol} \).
- Using the same temperature \( T \) and gas constant \( R \), we can write the equation for hydrogen:
\[ V_{RMS, H2} = \sqrt{\frac{3RT}{2 \times 10^{-3}}} \]

5. **Relating the Two Equations**:
Since \( 3RT \) is the same for both gases, we can set up the ratio:
\[ \frac{V_{RMS, N2}}{V_{RMS, H2}} = \sqrt{\frac{M_{H2}}{M_{N2}}} \]
Substituting the known values:
\[ \frac{500}{V_{RMS, H2}} = \sqrt{\frac{2 \times 10^{-3}}{28 \times 10^{-3}}} \]

6. **Calculating the Ratio**:
Simplifying the fraction:
\[ \frac{500}{V_{RMS, H2}} = \sqrt{\frac{2}{28}} = \sqrt{\frac{1}{14}} \approx 0.2673 \]

7. **Solving for \( V_{RMS, H2} \)**:
Rearranging gives:
\[ V_{RMS, H2} = \frac{500}{0.2673} \approx 1870.8 \, \text{m/s} \]

8. **Final Answer**:
Therefore, the RMS speed of hydrogen molecules at the same temperature is approximately:
\[ V_{RMS, H2} \approx 1870 \, \text{m/s} \]