A stainless steel chamber contains Ar gas at a temperature T and pressure P. The total number of Ar atoms in the chamber is n. Now Ar gas in the chamber is rep,aced by `CO_(2)` gas and the total number of `CO_(2)` molecules in the chamber is `n//2` at the same temperature T. The pressure in the chamber now is P'. Which one of the following relations relations holds true ? (Both the gases behave as ideal gases)

Options

A. P'=P
B. P'=2P
C. `P'=P//2`
D. `P' P//4`

Explanation

To solve the problem, we will use the ideal gas law, which states that:

\[ PV = nRT \]

Where:
- \( P \) = pressure of the gas
- \( V \) = volume of the gas
- \( n \) = number of moles of the gas
- \( R \) = universal gas constant
- \( T \) = temperature of the gas

### Step-by-Step Solution:

1. **Initial Conditions with Argon Gas:**
- Let the initial number of moles of Argon gas be \( n \).
- The initial pressure of Argon gas is \( P \).
- The ideal gas equation can be written as:
\[ P \cdot V = n \cdot R \cdot T \tag{1} \]

2. **Final Conditions with Carbon Dioxide Gas:**
- When Argon gas is replaced by Carbon Dioxide (CO₂), the number of CO₂ molecules in the chamber is \( \frac{n}{2} \).
- We need to find the new pressure \( P' \) when the number of moles of CO₂ is \( \frac{n}{2} \).
- The ideal gas equation for CO₂ can be written as:
\[ P' \cdot V = \left(\frac{n}{2}\right) \cdot R \cdot T \tag{2} \]

3. **Relating the Two Conditions:**
- From equation (1), we can express \( P \):
\[ P = \frac{nRT}{V} \tag{3} \]
- From equation (2), we can express \( P' \):
\[ P' = \frac{\left(\frac{n}{2}\right)RT}{V} = \frac{nRT}{2V} \tag{4} \]

4. **Finding the Relation Between \( P \) and \( P' \):**
- Now, we can substitute equation (3) into equation (4):
\[ P' = \frac{1}{2} \cdot P \tag{5} \]

5. **Conclusion:**
- Thus, the relation that holds true is:
\[ P' = \frac{P}{2} \]

### Final Answer:
The correct relation is \( P' = \frac{P}{2} \).