Two identical vessels contain helium and hydrogen gases at same temperature, then

Options

A. (a)average kinetic energy per mole of hydrogen = average kinetic energy per mole of helium
B. (b)average translational kinetic energy per mole of hydrogen = average translational kinetic energy per mole of helium
C. (c)average kinetic energy per mole of hydrogen `=(3)/(5)` average kinetic energy per mole of helium
D. (d)average kinetic energy per mole of hydrogen `=(5)/(3)` average kinetic energy per mole of helium

Answer

To solve the problem regarding the average kinetic energy of helium and hydrogen gases in identical vessels at the same temperature, we can follow these steps:

### Step 1: Understand the concept of average kinetic energy
The average kinetic energy of a gas is given by the formula:
\[ KE_{avg} = \frac{F}{2} RT \]
where \( F \) is the degrees of freedom, \( R \) is the universal gas constant, and \( T \) is the temperature in Kelvin.

### Step 2: Identify the type of gases
- Helium (He) is a monatomic gas, which means it has 3 degrees of freedom (translational motion only).
- Hydrogen (H₂) is a diatomic gas, which has 5 degrees of freedom (3 translational and 2 rotational).

### Step 3: Calculate the average kinetic energy for each gas
- For helium (monatomic):
\[ KE_{He} = \frac{3}{2} RT \]
- For hydrogen (diatomic):
\[ KE_{H2} = \frac{5}{2} RT \]

### Step 4: Find the ratio of average kinetic energies
To find the ratio of the average kinetic energies of hydrogen to helium:
\[ \frac{KE_{H2}}{KE_{He}} = \frac{\frac{5}{2} RT}{\frac{3}{2} RT} = \frac{5}{3} \]

### Step 5: Conclusion on average kinetic energy
From the calculations, we can conclude that the average kinetic energy of hydrogen is greater than that of helium by a ratio of \( \frac{5}{3} \).

### Step 6: Consider translational kinetic energy
The translational kinetic energy for an ideal gas can also be expressed as:
\[ KE_{trans} = \frac{3}{2} nRT \]
Since both gases are in identical vessels with the same amount of substance (n = 1 for both) and at the same temperature, their translational kinetic energies will be equal:
\[ KE_{trans, He} = KE_{trans, H2} = \frac{3}{2} RT \]

### Final Answer
- The ratio of average kinetic energies is \( \frac{5}{3} \) (for hydrogen to helium).
- The translational kinetic energies of both gases are equal.