Select a Van der Waals equation variable to calculate from the supplied molar quantities. Volume mode reports admissible model roots without selecting a phase.

Approximate molar equation of state: v = total volume ÷ moles. Volume mode reports admissible model roots; it does not determine phase equilibrium.

Universal molar R is supplied; change it only for a deliberate consistency check.

Related Calculators

Van Der Waals Equation Formula

The van der Waals equation is used to describe the behavior of real gases, taking into account intermolecular forces and the finite size of gas molecules. In molar form (using molar volume v = V/n), the equation is:

(P + a / v²)(v - b) = RT

Variables:

  • P is the pressure of the gas
  • v is the molar volume of the gas (v = V/n)
  • a is the van der Waals attraction parameter for the gas
  • b is the van der Waals co-volume (excluded volume) parameter for the gas
  • R is the ideal gas constant
  • T is the absolute temperature of the gas

Select the unknown variable, then enter the requested quantities and units. Use absolute pressure and temperature above absolute zero, nonnegative a and b, and v greater than b. Universal molar R = 8.31446261815324 J/(mol·K) is supplied. The R mode is a consistency check, not a way to change the physical constant.

What is a Van Der Waals Equation?

The van der Waals equation is a mathematical expression that describes the behavior of real gases, taking into account the finite size of molecules and the intermolecular forces between them. It is an improvement over the ideal gas law, which assumes that gas molecules have no volume and do not interact with each other. A common form (using total volume V and amount of substance n) is:

(P + a(n / V)²)(V - nb) = nRT

Dividing by n and using molar volume v = V/n gives the molar form used by the calculator: (P + a/v²)(v - b) = RT. The term a(n/V)² (or a/v² in molar form) corrects pressure for attractive forces between molecules, and the term nb (or b in molar form) accounts for the finite volume of the molecules by reducing the volume available to the gas.

How to Calculate Van Der Waals Equation?

The following steps outline how to calculate the Van Der Waals Equation:

  1. First, determine the values of the variables: pressure (P), temperature (T), and the van der Waals constants (a and b). Use the molar volume v (or convert from total volume using v = V/n).
  2. Next, substitute the values into the van der Waals equation (molar form): (P + a/v²)(v - b) = RT.
  3. Next, simplify the equation by expanding and rearranging terms.
  4. Finally, solve for the unknown. Volume mode solves the cubic v³ − (b + RT/P)v² + (a/P)v − ab/P = 0 and verifies admissible roots in the original equation. Multiple roots may exist. Positive isothermal slope is mechanically unstable; negative slope alone does not establish the equilibrium phase. No Maxwell construction or Gibbs comparison is performed.
  5. After obtaining the solution, check your answer by substituting the values back into the original equation.

Example Problem:

Use the following variables as an example problem to test your knowledge (molar form):

Pressure (P) ≈ 12.237 atm (the quantity to calculate)

Molar Volume (v) = 2 L/mol

Temperature (T) = 300 K

Van Der Waals constant (a) = 1.36 L^2 atm/mol^2

Van Der Waals constant (b) = 0.0427 L/mol. With R = 0.08205736608 atm·L/(mol·K), P = RT/(v − b) − a/v² ≈ 12.237 atm. The original 3 atm example was inconsistent with its other values. This is an approximate pure-fluid equation of state; no gas identity or phase-equilibrium data is supplied.