Question

A real gas is produced from a gas reservoir at a constant temperature of 30°C. The compressibility factor (\(Z\)) is observed to change with pressure (\(P\)) at a rate of \[ \left( \frac{\partial Z}{\partial P} \right)_T = Z^2 \] The difference in the compressibility of the real gas from the ideal gas at a given pressure (\(P\)) and temperature (\(T\)) is:

• A. \( Z \)
• B. \( Z^2 \)
• C. \( \sqrt{Z} \)
• D. \( \frac{1}{Z^2} \)

Hint:

The compressibility factor \(Z\) is used to quantify the deviation of a real gas from ideal gas behavior. For ideal gases, \(Z = 1\).

Answer 1

The Correct Option is A

The problem involves the compressibility factor (\(Z\)) of a real gas and its deviation from the ideal gas behavior. The given relation tells us how \(Z\) changes with pressure (\(P\)) at a constant temperature (\(T\)): \[ \left( \frac{\partial Z}{\partial P} \right)_T = Z^2 \] Step 1: Understanding the compressibility factor.
The compressibility factor \(Z\) is a measure of the deviation of a real gas from ideal gas behavior. For ideal gases, \(Z = 1\) at all pressures, but for real gases, \(Z\) can be greater than or less than 1 depending on the conditions.
Step 2: Analyze the given relation.
We are given that the rate of change of the compressibility factor with respect to pressure is proportional to \(Z^2\). This means that as pressure increases, the change in \(Z\) becomes more significant.
Step 3: Interpretation of the problem.
The difference in the compressibility of the real gas from the ideal gas is represented by \(Z\), which accounts for the deviation from ideal behavior. Hence, the correct answer is the compressibility factor itself, \(Z\).
Step 4: Conclusion.
Therefore, the correct answer is \(Z\).