If excluded volume is taken as zero, compressiblity factor Z is

Options

A. `(1 - (a)/(RTV))`
B. `(1+ (Pb)/(RT))`
C. `(PV)/(RT)`
D. `(RT)/(PV)`

Explanation

To find the compressibility factor \( Z \) when the excluded volume is taken as zero, we can follow these steps:

### Step 1: Understand the Compressibility Factor
The compressibility factor \( Z \) is defined as:
\[ Z = \frac{PV}{RT} \]
where \( P \) is the pressure, \( V \) is the volume, \( R \) is the ideal gas constant, and \( T \) is the temperature.

### Step 2: Consider the Van der Waals Equation
The Van der Waals equation for real gases is given by:
\[ \left(P + \frac{a n^2}{V^2}\right)(V - nb) = nRT \]
where \( a \) and \( b \) are constants specific to the gas.

### Step 3: Set Excluded Volume to Zero
If we take the excluded volume to be zero, we can assume \( b = 0 \). This simplifies the equation to:
\[ P + \frac{a n^2}{V^2} = \frac{nRT}{V} \]

### Step 4: Rearranging the Equation
For one mole of gas (\( n = 1 \)):
\[ P + \frac{a}{V^2} = \frac{RT}{V} \]
Rearranging gives:
\[ P = \frac{RT}{V} - \frac{a}{V^2} \]

### Step 5: Substitute into the Compressibility Factor Equation
Now, substituting \( P \) back into the equation for \( Z \):
\[ Z = \frac{PV}{RT} = \frac{\left(\frac{RT}{V} - \frac{a}{V^2}\right)V}{RT} \]
This simplifies to:
\[ Z = 1 - \frac{a}{RTV} \]

### Step 6: Conclusion
Thus, the compressibility factor \( Z \) when the excluded volume is taken as zero is:
\[ Z = 1 - \frac{a}{RTV} \]