The ratio of number of collision per second at the walls of containers by `H_(2)` and Ne gas molecules kept at same volume and temperature

Explanation

To find the ratio of the number of collisions per second at the walls of a container by hydrogen (H₂) and neon (Ne) gas molecules kept at the same volume and temperature, we can follow these steps:

### Step 1: Understand the formula for collision frequency
The number of collisions per second (Z) at the walls of a container can be expressed as:
\[ Z = 2 \pi V \sigma^2 n \]
where:
- \( V \) is the average velocity of the gas molecules,
- \( \sigma \) is the molecular diameter,
- \( n \) is the number of molecules.

### Step 2: Relate average velocity to temperature and molecular mass
The average velocity \( V \) of a gas molecule can be calculated using the formula:
\[ V = \sqrt{\frac{8RT}{\pi m}} \]
where:
- \( R \) is the universal gas constant,
- \( T \) is the absolute temperature,
- \( m \) is the molecular mass of the gas.

### Step 3: Substitute average velocity into the collision frequency formula
Substituting the expression for average velocity into the collision frequency formula gives:
\[ Z \propto \frac{1}{\sqrt{m}} \]
This indicates that the number of collisions is inversely proportional to the square root of the molecular mass.

### Step 4: Set up the ratio of collisions for H₂ and Ne
To find the ratio of the number of collisions per second for H₂ and Ne, we can write:
\[ \frac{Z_{H_2}}{Z_{Ne}} = \frac{\sqrt{m_{Ne}}}{\sqrt{m_{H_2}}} \]

### Step 5: Insert the molecular masses
The molecular mass of H₂ is approximately 2 g/mol, and the molecular mass of Ne is approximately 20 g/mol. Therefore, we have:
\[ \frac{Z_{H_2}}{Z_{Ne}} = \frac{\sqrt{20}}{\sqrt{2}} \]

### Step 6: Simplify the ratio
Now, simplifying the ratio:
\[ \frac{Z_{H_2}}{Z_{Ne}} = \frac{\sqrt{20}}{\sqrt{2}} = \sqrt{\frac{20}{2}} = \sqrt{10} \]

### Final Answer
Thus, the ratio of the number of collisions per second at the walls of the container by H₂ and Ne is:
\[ \frac{Z_{H_2}}{Z_{Ne}} = \sqrt{10} \]

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