# How to evaluate the critical exponents of modified van der Waals equation?

The given modified van der Waals equation is $$(P+(a/v)^{n})(v-b)=RT$$ where $(n>1)$. What is the physical significance of the power $n$ in the above equation. How could one evaluate the critical constants and exponents for this modified equation ? Suggest some reference to read this

• Also, $n$ just might have no physical significance but a mathematical significance that as you increase the number of moles $n$ of a gas, the correction term for pressure increases by a factor of $\frac{an^2}{V^2}$,suggesting that the pressure of larger number of gas molecules is larger than for the smaller number of molecules, which hence helps us to arrive at the physical significance of $n$ – Zlatan May 13 '17 at 2:25

## 1 Answer

The modified Van Der Walls Equation is

$(P+\frac{an^2}{V^2})(V-nb)=nRT$......... $(1)$

where $a$ is a Van Der Walls constant whose value depends upon intramolecular forces of attraction within the gas. $b$ is the correction term for Volume of gas.

Putting $n=1$ in $(1)$,

$(P+\frac{a}{V^2})(V-b)=RT$......... $(1)$ or

$V^3-(\frac{RT}{P}+b)V^2+\frac{a}{P}V-\frac{ab}{P}=0$.......... $(2)$

This equation has three real roots. Also, at critical condition

All three roots are equal

So, $V_1=V_2=V_3=V_c$ or

$(V-V_c)^3=0$ or

$V^3+3VV_{c}^2-3V^2V_{c}-V_{c}^3=0$.........$(3)$

Now, on comparing the coefficients of $(2)$ and $(3)$,you get

$V_{c}^3=\frac{ab}{P_c}$,

$\frac{RT_c}{P_c}+b=3V_c$ and

$\frac{a}{P_c}=3V_{c}^2$

Solving,

$V_c=3b$

$P_c=\frac{a}{27b^2}$

$T_c=\frac{8a}{27Rb}$

where $V_c, P_c$ and $T_c$ are critical Volume, Pressure and Temperature respectively.

Hope this helps, brother.

• Here , in the solution, it is assumed that Volume is squared instead of taking it as (V^n) as pressure correction. How could one solve it in general ?? – user135580 May 13 '17 at 13:40
• for any power $n$,one could still multiply and expand all terms and then apply the same condition, i.e. The roots are equal at critical point. Therefore, you will get the same critical values for pressure, temperature and volume. – Zlatan May 14 '17 at 4:01
• Zlatan, Thank you so much for the explanation. Yet , I don't understand why the roots are always equal in the critical condition whether it is cubic equation or volume or nth root of volume (V^3 or V^n) – user135580 May 14 '17 at 6:45
• @user135580 it's the necessary criterion for critical condition. Because if the roots wouldn't be equal, we would get different values for critical pressure, volume and temperature, but as the word suggests "critical" should be one fixed value. – Zlatan May 14 '17 at 10:56
• Glad I could help ☺ – Zlatan May 14 '17 at 10:56