Van der Waals Equation Calculator

Created by Dominik Czernia, PhD
Reviewed by Bogna Szyk and Steven Wooding
Last updated: Oct 28, 2022

This van der Waals equation calculator is a handy tool that allows you to analyze the physical properties of real gases. It uses the van der Waals equation, which is a thermodynamic equation of state that improves the ideal gas law equation. Have you ever wondered what is a real gas? Read on to learn about the real gas definition and the real gas law.

Do you need to analyze the gas state equation but don't know which one you should use? You have to choose the correct theory: ideal gas vs. real gases. The following text explains the difference between an ideal and a real gas. You will find out the advantages of using the van der Waals equation of state instead of an ideal gas equation.

Van der Waals introduced two parameters (so-called van der Waals parameters), which are related to the critical point of gas, i.e., to the point at which it is impossible to distinguish between liquid and vapor (liquid and its vapor can coexist). In the text below, we have also written about the meaning of these constants and how you can estimate them.

Ideal gas vs. real gas

An ideal gas consists of a large number of randomly moving point particles, which can collide with each other and the walls of the container. It is a so-called kinetic molecular theory, which we have described in our thermal energy calculator. You may also be interested in our particle velocity calculator, where you can find the average velocity of particles in a gas at a given temperature.

The real gas law, a modification of the ideal gas law, takes into account two additional aspects:

  1. Molecules do not only collide with each other like in an ideal gas but can also attract each other within a distance of several molecules' radii. Because there are no gas particles behind the surface of the container, molecules near the surface are attracted to the material. As a result, the actual real gas pressure on the container walls is less than it would be in an ideal gas (see pressure calculator).

  2. Molecules are particles with a non-zero volume – not material points, like in an ideal gas. Therefore, the volume of a whole gas can't be less than a certain constant.

Van der Waals constants

Van der Waals constants are substance-specific constants that can be calculated using the parameters of the critical point: pressure PcP_{\rm c}, temperature TcT_{\rm c}, and the molar volume VcV_{\rm c}. At a critical point, both the liquid and gas phases of a substance have the same density – they are indistinguishable. There is a useful relation between critical point parameters:

8PcVc=3RTc,8P_{\rm c}V_{\rm c} = 3R\,T_{\rm c},

where RR is the gas constant R=8.3144598 J/(mol ⁣ ⁣K)\small R = 8.3144598\ \rm J / (mol\! \cdot\! K).

In the real gas law, one of the van der Waals constants is called attraction parameter aa, which takes into account that particles can attract each other, and the second is repulsion parameter bb, which is the effective molecular volume (particles are not material points).

Our van der Waals equation calculator uses the following formulas for the van der Waals constants:

a=3PcVc2andb=Vc/3a = 3P_cV_c^2\quad \text{and}\quad b = V_c/3

If the interaction between molecules (a=0)\small(a = 0) and the size of the molecules (b=0)\small(b = 0) can be neglected, i.e., if we can treat gas as the ideal gas, then the van der Waals equation of state transforms into the ideal gas equation.

Van der Waals equation of state

Our van der Waals equation calculator is divided into two parts. First, you should specify the critical parameters of the considered gas to estimate van der Waals constants. You can change those constants directly in the advanced mode or choose one of the typical gases.

In the second part of the van der Waals equation calculator, you can find what is the relation between volume, pressure, and temperature in real gas with the following relation:

(P+an2V2)(Vnb)=nRT,\left(P + a\frac{n^2}{V^2}\right)(V - nb) = nR\,T,

where:

  • PP – Pressure of the gas;
  • VV – Volume of the gas;
  • TT – Temperature of the gas;
  • nn – Number of moles of gas (see molarity calculator); and
  • aa and bb - Van der Waals parameters (see the previous section).

The constant bb value corresponds to the volume of one mole of the molecules (it is a correction for finite molecular size). Therefore, to avoid a situation where the volume of the molecules is greater than that of the whole gas, we must impose a condition V/n>b\small V / n > b.

The van der Waals equation is generally a very good approximation of the real gas state equation, especially for high pressures and under temperature and pressure conditions close to the gas condensation parameters.

Dominik Czernia, PhD
Critical point
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