The compressibility factor (Z) is defined as the ratio of the product of pressure and volume for a real gas to that of an ideal gas under the same conditions: $Z=\frac{PV}{RT}$. For an ideal gas, Z = 1. The van der Waals equation accounts for real gas behavior and is written as: $(P+\frac{a}{V^2})(V-b)=RT$, where 'a' corrects for intermolecular attractions and 'b' for the finite volume of gas molecules.

At low pressure, the volume is large, so V >> b. This allows us to approximate (V - b) ≈ V. The equation simplifies to: $(P+\frac{a}{V^2})V=RT$.

Expanding this: $PV+\frac{a}{V}=RT$.

Rearranging for PV: $PV=RT-\frac{a}{V}$.

Now, substitute into the compressibility factor definition: $Z=\frac{PV}{RT}=\frac{RT-\frac{a}{V}}{RT}=1-\frac{a}{VRT}$.

Since pressure is low, volume V is large, and we can use the ideal gas law approximation V = RT/P. Substituting this in: $Z=1-\frac{a}{\left(\frac{RT}{P}\right)RT}=1-\frac{aP}{{\left(RT\right)}^2}$. However, the more standard and directly derived form from the previous step is $Z=1-\frac{a}{VRT}$, which matches the second option.

Therefore, the correct expression for the compressibility factor at low pressure is: $Z=1-\frac{a}{VRT}$.

Related Topics and Formulae

Van der Waals Equation: $(P+\frac{a}{n^2}\frac{1}{V^2})(V-nb)=nRT$ (for n moles). Constants 'a' and 'b' are specific to each gas and account for attraction and volume, respectively.

Compressibility Factor (Z): A dimensionless quantity that measures the deviation from ideal gas behavior. Z = 1 for ideal gases; Z < 1 indicates attractive forces dominate; Z > 1 indicates repulsive forces dominate.

Low-Pressure Approximation: At low pressure, gas molecules are far apart, so volume correction (b) is negligible, but attraction (a) is significant, leading to Z < 1.