What are the different ways of increasing the number of molecules collisions per unit time against the walls of the vessel containing a gas ?

Answer

To solve the question of how to increase the number of molecular collisions per unit time against the walls of a vessel containing a gas, we can follow these steps:

### Step 1: Understand the Relationship
The number of collisions per unit time (Z) can be expressed with the formula:
\[ Z \propto n \cdot v_{\text{rms}} \]
where:
- \( n \) = number of moles per unit volume (moles/volume)
- \( v_{\text{rms}} \) = root mean square velocity of gas molecules

### Step 2: Increase the Number of Moles
To increase the number of collisions, we can increase the number of moles of gas in the vessel. This can be achieved by:
- Adding more gas to the container.
- Increasing the concentration of the gas.

### Step 3: Decrease the Volume
The number of collisions is inversely proportional to the volume of the container. Thus, to increase the number of collisions:
- Decrease the volume of the gas container. This can be done by compressing the gas, which forces the molecules closer together, increasing the frequency of collisions.

### Step 4: Increase the Root Mean Square Velocity
The root mean square velocity \( v_{\text{rms}} \) is influenced by temperature. The formula for \( v_{\text{rms}} \) is:
\[ v_{\text{rms}} = \sqrt{\frac{3RT}{M}} \]
where:
- \( R \) = universal gas constant
- \( T \) = absolute temperature
- \( M \) = molar mass of the gas

To increase \( v_{\text{rms}} \):
- Increase the temperature of the gas. Higher temperatures give gas molecules more kinetic energy, resulting in higher speeds and more frequent collisions.

### Conclusion
To summarize, the different ways to increase the number of molecular collisions per unit time against the walls of a gas container are:
1. **Increase the number of moles of gas** in the container.
2. **Decrease the volume** of the container.
3. **Increase the temperature** of the gas to raise the root mean square velocity of the molecules.