If the collision frequency of a gas at 1 atm pressure is Z, then its collision frequency at 0.5 atm is

Options

A. 1.0 Z
B. 0.25 Z
C. 2Z
D. 0.5 Z

Explanation

To solve the problem, we need to determine the collision frequency of a gas at a pressure of 0.5 atm, given that its collision frequency at 1 atm is Z.

### Step-by-Step Solution:

1. **Understand the Relationship**:
The collision frequency (Z) of a gas is directly proportional to the square of the pressure (P) at constant temperature. This can be expressed as:
\[ Z \propto P^2 \]

2. **Set Up the Proportionality**:
Let \( Z_1 \) be the collision frequency at 1 atm (which is given as Z) and \( Z_2 \) be the collision frequency at 0.5 atm. The pressures are:
- \( P_1 = 1 \, \text{atm} \)
- \( P_2 = 0.5 \, \text{atm} \)

Using the proportionality, we can write:
\[ \frac{Z_1}{Z_2} = \frac{P_1^2}{P_2^2} \]

3. **Substitute Known Values**:
Substitute \( Z_1 = Z \), \( P_1 = 1 \, \text{atm} \), and \( P_2 = 0.5 \, \text{atm} \) into the equation:
\[ \frac{Z}{Z_2} = \frac{(1)^2}{(0.5)^2} \]

4. **Calculate the Right Side**:
Calculate \( (0.5)^2 \):
\[ (0.5)^2 = 0.25 \]
Thus, the equation becomes:
\[ \frac{Z}{Z_2} = \frac{1}{0.25} = 4 \]

5. **Solve for \( Z_2 \)**:
Rearranging the equation gives:
\[ Z_2 = \frac{Z}{4} \]

6. **Conclusion**:
Therefore, the collision frequency at 0.5 atm is:
\[ Z_2 = 0.25Z \]

### Final Answer:
The collision frequency at 0.5 atm is \( 0.25Z \).