Compressibility factor for `H_(2)` behaving as real gas is

Options

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

Answer

To determine the compressibility factor (Z) for hydrogen gas (H₂) behaving as a real gas, we will use the Van der Waals equation. Here’s the step-by-step solution:

### Step 1: Understand the Van der Waals Equation
The Van der Waals equation for real gases is given by:
\[ \left(P + \frac{a}{V^2}\right)(V - b) = RT \]
where:
- \( P \) = pressure of the gas
- \( V \) = volume of the gas
- \( T \) = temperature
- \( R \) = universal gas constant
- \( a \) and \( b \) are Van der Waals constants specific to the gas.

### Step 2: Rearranging the Equation
For high pressures, the term \( \frac{a}{V^2} \) becomes negligible compared to \( P \). Thus, we can simplify the equation to:
\[ PV - Pb = RT \]

### Step 3: Isolate the Compressibility Factor
The compressibility factor \( Z \) is defined as:
\[ Z = \frac{PV}{RT} \]
From the rearranged Van der Waals equation, we can express \( PV \) in terms of \( RT \) and \( Pb \):
\[ PV = RT + Pb \]

### Step 4: Substitute into the Compressibility Factor Equation
Now, substituting \( PV \) into the equation for \( Z \):
\[ Z = \frac{RT + Pb}{RT} \]

### Step 5: Simplify the Expression
This can be simplified to:
\[ Z = 1 + \frac{Pb}{RT} \]

### Conclusion
Thus, the compressibility factor \( Z \) for hydrogen gas (H₂) behaving as a real gas at high pressure is:
\[ Z = 1 + \frac{Pb}{RT} \]