The root mean square speed of an ideal gas is given by : `u_("rms") = sqrt((3RT)/M)` Thus we conclude that `u_("rms")` speed of the ideal gas molecules is proportional to square root of the temperature and inversely proportional to the square root of the molar mass. The translational kinetic energy per mole can also be given as `1/2Mu_(rms)^(2)` . The mean free path (`lambda`) is the average of distances travelled by molecules in between two successive collisions whereas collision frequency (C.F.) is expressed as number of collisions taking place in unit time. The two terms `lambda` and C.F. are related by : `C.F = (u_("rms")/lambda)`
A jar contains He and H, in the molar ratio 1 : 5. The ratio of mean translational kinetic energy at the same temperature is

Options

A. `1:5`
B. `5:1`
C. `2:1`
D. `1:1`

Answer

To solve the problem, we need to find the ratio of mean translational kinetic energy of helium (He) and hydrogen (H) at the same temperature, given that they are in a molar ratio of 1:5.

### Step 1: Understand the formula for translational kinetic energy
The mean translational kinetic energy (E) of an ideal gas is given by the formula:
\[ E = \frac{3}{2} RT \]
where:
- \( R \) is the universal gas constant,
- \( T \) is the absolute temperature.

### Step 2: Determine the translational kinetic energy for He and H
Since the question states that the gases are at the same temperature, we can denote the kinetic energy for helium and hydrogen as follows:
- For helium (He):
\[ E_{He} = \frac{3}{2} RT \]
- For hydrogen (H):
\[ E_{H} = \frac{3}{2} RT \]

### Step 3: Calculate the ratio of kinetic energies
Since both gases are at the same temperature, their kinetic energies are equal. Therefore, the ratio of their kinetic energies is:
\[ \frac{E_{He}}{E_{H}} = \frac{\frac{3}{2} RT}{\frac{3}{2} RT} = 1 \]

### Step 4: Conclusion
Thus, the ratio of mean translational kinetic energy of helium to hydrogen at the same temperature is:
\[ \text{Ratio} = 1:1 \]

### Final Answer
The ratio of mean translational kinetic energy at the same temperature for helium and hydrogen is \( 1:1 \).

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