What is compressibility factor $(Z)$? Explain $(i)$ deviation factor $(Z = 1)$,$(ii)$ $Z > 1$,$(iii)$ $Z < 1$,$(iv)$ deviation graph,and $(v)$ the relation between molar volume and $Z$.

(A) Compressibility factor $(Z)$ is defined as the ratio of the product of pressure and volume to the product of the number of moles,gas constant,and temperature: $Z = \frac{pV}{nRT}$.
$(i)$ For an ideal gas,$Z = 1$ at all temperatures and pressures,as it follows the equation $pV = nRT$. On a $Z$ vs $p$ graph,this is represented by a horizontal line parallel to the pressure axis.
$(ii)$ $Z > 1$ (Positive deviation): This occurs at high pressures where real gases are less compressible than ideal gases. Gases like $H_2$ and $He$ show $Z > 1$ at all pressures because intermolecular forces are negligible compared to the volume occupied by molecules.
$(iii)$ $Z < 1$ (Negative deviation): This occurs at intermediate pressures where attractive forces dominate,making the gas more compressible than an ideal gas. Gases like $CH_4$ and $CO_2$ show this behavior at lower pressures.
$(iv)$ The deviation graph plots $Z$ on the $y$-axis against $p$ on the $x$-axis. The ideal gas line is a horizontal line at $Z = 1$. Real gases show curves that deviate from this line depending on pressure and temperature.
$(v)$ The relation between molar volume and $Z$ is given by $Z = \frac{V_{real}}{V_{ideal}}$,where $V_{real}$ is the actual molar volume and $V_{ideal} = \frac{RT}{p}$ is the molar volume of an ideal gas at the same temperature and pressure.