Explanation of Behaviour of Real Gases Using Vander Waal's Equation

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Explanation of Behaviour of Real Gases Using Vander Waal’s Equation

As we know the Van der Walls equation for n moles of a real gas is,

\( \left( {P + \frac{{an^2 }}{{V^2 }}} \right)\left( {V – bn} \right) = nRT \)

For one mole of gas, put n=1,

\( \left( {P + \frac{{a}}{{V^2 }}} \right)\left( {V – b} \right) = RT \tag{1}\\ \)

At very low Pressure :

At very low pressure, the volume V of the gas is very large.
As we know, the VWL constants a and b are very small.
i.e., V>>>a ⇒ \( \frac{a}{V^2} \) can be neglected
and V>>>b ⇒ V-b \(\approx\) V∴ Equation 1 reduces to
\(\fbox{PV = RT}\) , viz., the ideal gas equation.
Thus, at extremely low pressures, the real gas obeys ideal gas equation.

At moderate Pressure :

At moderate pressure, the volume decreases;
⇒\( \frac{a}{V^2} \) becomes appreciable and hence it can not be neglected.
Since V is sufficiently larger than b,
i.e., V>>b ⇒ V-b \(\approx\) V
Equation 1 reduces to
\((P+\frac{a}{V^2})V=RT\)

⇒\(PV+\frac{a}{V}=RT\)
⇒\(PV=RT-\frac{a}{V}\)
⇒\(\frac{PV}{RT}=1-\frac{a}{VRT}\)
⇒\(\fbox{z=1-\(\frac{a}{VRT}\)}\)
⇒z < 1 i.e., real gases show -ve deviations.
∴ V_real < V_ideal
⇒gas is more compressible than that expected from ideal behaviour.
i.e.,attractive forces are predominant.

At High Pressure :

When the pressure is very high, the volume V becomes extremely small.
So the correction term b can not be neglected as compared to V.
Since, P>> \( \frac{a}{V^2} \) ;
so, we can neglect the term \( \frac{a}{V^2} \) as compared to P.
∴ Equation 1 reduces to
\(P(V-b)=RT\)
Dividing both sides by RT,
\( \frac{PV}{RT} = 1+ \frac{Pb}{RT}\)
\(\fbox{z= 1+\(\frac{Pb}{RT}\)}\)
As the pressure increases, the term Pb in equation increases hence the value of z>1
z > 1 i.e., real gases show +ve deviations.
⇒V_real > V_ideal
⇒gas is less compressible than that expected from ideal behaviour.
i.e.,Repulsive forces are predominant.

At High Temperature :

If the temperature is sufficiently high at a given pressure then the volume V becomes considerably large so that P >> \( \frac{a}{V^2} \)
i.e., we can neglect \( \frac{a}{V^2} \) in comparison to P.
Also, V >> b ,i.e., we can neglect b in comparison to large volume V.
Equation 1 reduces to,
\(\fbox{PV = RT}\)
Hence real gases behaves nearly ideally at high temperature.

At Very Low Pressures & Very High Temperatures :

Under the conditions of very low pressure and very high temperatures, the Volume V becomes so large.
So \( \frac{a}{V^2} \) is very small and we can neglect this term.
Also V>>b, so we can neglect b
∴ Equation 1 reduces to,
\(\fbox{PV = RT}\)
Hence, Under these conditions, the real gas approaches ideal behaviour.

Exceptional Behaviour of H₂ & He

As shown in fig., H₂ & He shows exceptional behaviour because for these gases the compressibility factor z always increase with increase in pressure.
As hydrogen and helium are very small masses
∴ Attractive forces between their molecules are negligible
Hence, the term a for H2 & He extremely small
∴ \( \frac{a}{V^2} \) is negligible at all pressures and at ordinary temperatures.
∴ Equation 1 reduces to,
PV= RT + Pb
Dividing both sides by RT,
\( \frac{Pv}{RT} \) = 1+ \( \frac{Pb}{RT} \)
\(\fbox{z= 1+\(\frac{Pb}{RT}\)}\)
⇒ z > 1
⇒z vs P curve for H2 and He always lies above the ideal gas curve and shows upward trend with increase in pressure
Hence H2 & He both always show +ve deviation which increase with increase in pressure.