Question

The compressibility factor is always greater than 1 and increases with increase in pressure for hydrogen and helium. (True or False)

Answer 1

Hint: The compressibility factor represents the deviation from ideal behaviour exhibited by real gases. This deviation arises because the intermolecular forces are not negligible as previously assumed in kinetic theory of gases. This question can be answered if we know how the intermolecular forces behave at different pressures assuming temperature to be constant.

Complete Step by Step Solution:
The equation of state\[PV = nRT\]derived from the kinetic theory of gases is obeyed by ideal gases only. This “ideal gas equation” is valid only approximately for all real gases and that too under low pressures and high temperatures. Increasing the pressure and/or lowering the temperature will increase the deviations of real gases from the ideal gas behaviour.
These deviations are, mathematically best represented in terms of the compressibility factor (\[Z\]). The compressibility factor is defined as \[Z = \dfrac{{P{V_{real}}}}{{P{V_{ideal}}}} = \dfrac{{P{V_{real}}}}{{nRT}}\] \[ \Rightarrow Z = \dfrac{{{V_{real}}}}{{\left( {\dfrac{{nRT}}{P}} \right)}}\] \[ \Rightarrow Z = \dfrac{{{V_{real}}}}{{{V_{ideal}}}}\]
where \[P = \]pressure
            \[n = \]number of moles
            \[{V_{real}}\]= volume occupied by the real gas,
            \[{V_{ideal}}\]= volume occupied by the real gas if it behaved ideally
            \[R = \]Gas constant
            \[T = \]temperature

For ideal gases\[Z = 1\]at all temperatures and pressures. All real gases have Z close to unity at extremely low pressures. At high pressures, Z > 1 for real gases indicates that they are now less compressible than an ideal gas.
The cause for these deviations from ideal behaviour is an incorrect assumption in the kinetic theory of gases. That assumption is considering all intermolecular forces of attraction to be negligible. This assumption holds only at low pressures because the gas molecules lie very far apart from each other. At high enough pressures, the gas molecules some so close to each other that the intermolecular forces of attraction become so significant that they cannot be neglected anymore.

While for gases like carbon monoxide and ammonia, Z at first decreases below 1 at moderate pressures and then increases to above 1 at high pressures, for gases like hydrogen and helium, Z remains greater than 1 even at moderate pressures and only keeps increasing as the pressure is increased. This is because, while long-range attractive forces dominate at moderate pressures for carbon monoxide and ammonia, they are very weak for hydrogen and helium due to their small molecular sizes. Because of this, the real volume occupied by hydrogen and helium (\[{V_{real}}\]) is always greater than the ideal volume (\[{V_{ideal}}\]). Since \[{V_{real}} > {V_{ideal}} \Rightarrow Z\left( { = \dfrac{{{V_{real}}}}{{{V_{ideal}}}}} \right) > 1\].

Thus, the answer is True

Note: The above-mentioned behaviour of hydrogen and helium is true at\[0{\rm{ }}^\circ C\]. However, if the temperature is reduced to below \[ - 48{\rm{ }}^\circ {\rm{C}}\] for hydrogen and below\[ - 242{\rm{ }}^\circ {\rm{C}}\]for helium, their Z values will also drop below 1 at moderate pressures. Therefore, it is essential to keep the temperature in mind as well.