The compressibility factor (Z) is a measure of how much a real gas deviates from ideal gas behavior. It is defined as:

$Z=\frac{PV}{nRT}$

Given:

• Volume of gas A is twice that of gas B: $V_A=2V_B$
• Compressibility factor of A is thrice that of B: $Z_A=3Z_B$
• Equal number of moles: $n_A=n_B=n$
• Same temperature: $T_A=T_B=T$

Step 1: Write the compressibility factor equations for both gases.

For gas A: $Z_A=\frac{P_A{V}_A}{nRT}$

For gas B: $Z_B=\frac{P_B{V}_B}{nRT}$

Step 2: Substitute the given relationships.

Since $Z_A=3Z_B$, we can write:

$\frac{P_A{V}_A}{nRT}=3\times \frac{P_B{V}_B}{nRT}$

Step 3: Simplify the equation by canceling common terms (n, R, T are same for both gases).

$P_A{V}_A=3P_B{V}_B$

Step 4: Substitute $V_A=2V_B$ into the equation.

$P_A\times 2V_B=3P_B{V}_B$

Step 5: Cancel $V_B$ from both sides (assuming $V_B\ne 0$).

$2P_A=3P_B$

Step 6: Rearrange to find the relationship between $P_A$ and $P_B$.

$2P_A=3P_B\Rightarrow P_A=\frac{3}{2}{P}_B$

This can also be written as $2P_A=3P_B$, which matches the option: $2P_A=3P_B$.

Final Answer: $2P_A=3P_B$

Related Topics and Formulae

Compressibility Factor (Z): It indicates deviation from ideal gas behavior. For an ideal gas, Z=1. For real gases, Z can be less than 1 (attractive forces dominate) or greater than 1 (repulsive forces dominate). The formula is $Z=\frac{PV}{nRT}$.

Ideal Gas Law: The foundation for understanding gas behavior is the ideal gas equation: $PV=nRT$, which assumes no intermolecular forces and negligible molecular volume.

Real Gases: Real gases deviate from ideality due to intermolecular forces and the finite size of molecules. The van der Waals equation is a common modification to account for these: $(P+\frac{a{n}^2}{V^2})(V-nb)=nRT$, where 'a' and 'b' are constants specific to each gas.