# What is the value of isothermal compressibility of a van der Waals gas at critical point?

For a van der Waals gas, isothermal compressibility is $$\kappa=\frac{V^2(V-nb)^2}{nRTV^3-2an^2(V-nb)^2}$$ If one substitutes the values of critical temperature and volume in this formula, isothermal compressibility goes to infinity. What is the physical meaning of this? Is it right to use this equation at critical point?

• Of course it goes to infinity. What else did you expect? At critical point we have ${\partial{\mathfrak p}\over\partial V}=0$; that's the definition. – Ivan Neretin May 28 at 7:44
• Generally, the vdW equation is better to be avoided near critical point. Particularly, some partial derivatives of state variables go to infinity at the critical point. – Poutnik May 28 at 7:45
• Indeed, the vdW equation becomes less and less reliable near critical point. This particular detail, however, is reliable. – Ivan Neretin May 28 at 7:54
• I agree that a critical point is a point of inflection. If one has to compress at critical point at constant temperature, one has to move along the critical isotherm. How does this shed a physical meaning to $$\kappa\rightarrow\infty$$? – Nakul Aggarwal May 28 at 17:17