The mean speed of the molecules of a hydrogen sample equals the mean speed of the molecules of a helium sample. Calculate the ratio of the temperature of the hydrogen sample to the temperature of the helium sample.

Explanation

To solve the problem of finding the ratio of the temperature of a hydrogen sample (T1) to the temperature of a helium sample (T2) given that their mean speeds are equal, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Given Information**:
- The mean speed of hydrogen (V_avg_H2) is equal to the mean speed of helium (V_avg_He).
- We need to find the ratio T1/T2.

2. **Recall the Formula for Mean Speed**:
- The mean speed of a gas is given by the formula:
\[ V_{avg} = \sqrt{\frac{8RT}{\pi m}} \]
where:
- \( R \) is the universal gas constant,
- \( T \) is the temperature of the gas,
- \( m \) is the molar mass of the gas.

3. **Set Up the Equation**:
- For hydrogen:
\[ V_{avg, H2} = \sqrt{\frac{8RT_1}{\pi m_{H2}}} \]
- For helium:
\[ V_{avg, He} = \sqrt{\frac{8RT_2}{\pi m_{He}}} \]
- Since \( V_{avg, H2} = V_{avg, He} \), we can equate the two expressions:
\[ \sqrt{\frac{8RT_1}{\pi m_{H2}}} = \sqrt{\frac{8RT_2}{\pi m_{He}}} \]

4. **Simplify the Equation**:
- Squaring both sides to eliminate the square root gives:
\[ \frac{8RT_1}{\pi m_{H2}} = \frac{8RT_2}{\pi m_{He}} \]
- The \( 8R \) and \( \pi \) terms cancel out:
\[ \frac{T_1}{m_{H2}} = \frac{T_2}{m_{He}} \]

5. **Rearranging for Temperature Ratio**:
- Rearranging the equation gives:
\[ \frac{T_1}{T_2} = \frac{m_{H2}}{m_{He}} \]

6. **Substituting Molar Masses**:
- The molar mass of hydrogen (H2) is 2 g/mol and the molar mass of helium (He) is 4 g/mol:
\[ \frac{T_1}{T_2} = \frac{2}{4} = \frac{1}{2} \]

7. **Final Result**:
- Therefore, the ratio of the temperature of hydrogen to the temperature of helium is:
\[ T_1 : T_2 = 1 : 2 \]