If a gas gets half compressed, compared to an ideal gas, the compressibility factor Z is equal to

Options

A. 1
B. 2
C. `1/2`
D. none of these

Answer

To solve the problem of finding the compressibility factor \( Z \) when a gas is half compressed compared to an ideal gas, we can follow these steps:

### Step 1: Understand the Compressibility Factor
The compressibility factor \( Z \) is defined as:
\[ Z = \frac{V_m^{real}}{V_m^{ideal}} \]
where \( V_m^{real} \) is the molar volume of the real gas and \( V_m^{ideal} \) is the molar volume of the ideal gas.

### Step 2: Analyze the Compression
If a gas is "half compressed," it means that the volume of the gas has decreased to half of its original volume. Therefore, if we denote the original volume as \( V \), after half compression, the volume becomes:
\[ V_{compressed} = \frac{V}{2} \]

### Step 3: Relate the Volumes
For an ideal gas, the volume decreases with compression. However, since we are considering the compressibility factor, we need to relate the volumes of the real gas and the ideal gas under the same conditions.

When the gas is half compressed, the volume of the real gas \( V_m^{real} \) can be considered to be double that of the ideal gas volume under the same conditions. Thus:
\[ V_m^{real} = 2 \cdot V_m^{ideal} \]

### Step 4: Calculate the Compressibility Factor
Now, substituting this relationship into the formula for \( Z \):
\[ Z = \frac{V_m^{real}}{V_m^{ideal}} = \frac{2 \cdot V_m^{ideal}}{V_m^{ideal}} = 2 \]

### Conclusion
Thus, the compressibility factor \( Z \) when a gas is half compressed compared to an ideal gas is:
\[ Z = 2 \]