**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 H**_{2} & He

_{2}& He

**As shown in fig., H**_{2}& 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.**