**The Isotherms of van der Wall’s Equation **

**As we know, the Vanderwaals equation for one mole of gas is,**

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

**⇒****\(PV+\frac{a}{V}-Pb-\frac{ab}{V^2}=RT\)**

**Multiplying both sides by \(\frac{V^2}{P}\) and arranging the obtained equation in decreasing powers of V, we get**

**\(\fbox{\(V^3-(b+\frac{RT}{P})V^2 +\frac{aV}{P} -\frac{ab}{P}=0\)}\)**

**viz., a cubic equation in V, i.e.,For any fixed value of T & P, we have 3 values of V.**

**All three values may be real or only one may be real and other two imaginary.**

**To calculate the different values of V at different pressures (P) at a given temperature, we have to substitute the appropriate values of van der Walls constants a and b in above equation.**

**Now, these values of V are plotted against pressure P for a constant temperature T and we get various isotherms of Van der Wall’s equation:**

**Let us discuss the theoretical P-V isotherms of carbon dioxide according to Van der Walls equation:**

** ISOTHERM- I **

**At 13.1°C, we obtain a wave like portion KLMNOQ as shown in the fig.****If we decrease the volume , then pressure increases till point M and we obtain the curve KLM****Now on further decreasing the volume, the pressure also decreases till point O.**

**i.e., we obtain a curve MNO viz., not possible practically.****Now on further decrease in volume results in sudden increase in the pressure.**

__Note:__

**• These isomers resemble with the Andrew’s isotherms obtained Experimentally.**

**• Comparing this graph to that of real gases(Andrew’s experiment), we observe the difference that the horizontal line in case of real gases was replaced by the wavelike portion KLMNOQ.**

**• Small portions corresponding to curves LM and OQ can be realized in practice too.**

**• The curve LM represents super-saturated vapour & the curve OQ represents the superheated liquid.**

** ISOTHERM- II **

**Further as temperature is increased (25°C), the curved portions becomes smaller and smaller till critical temperature(31°C).And the three values of V get closer to one another. **

** ISOTHERM- III**

**At critical temperature(31°C), the curved portion disappears and only a point X is obtained.**

**At critical temperature, all three values of V become identical and equal to \({V}_C\).**

**At critical temperature, the three roots of Van der Walls equation are equal and the value of V represents the critical volume of the gas.**

** ISOTHERM- IV **

**Above critical temperature, isotherms indicate only one real value of V for any value of P.**

__Note::__Below critical temperature, a part of isotherms appears as a horizontal line.

**Along this line, volume changes at constant pressure P and we have 3 values of V marked as L, N, and Q. (Isotherm I)**