# Ideal Gas vs Non-Ideal Gas Example Problem

## Van Der Waals Equation Example Problem At low temperatures, real gases behave as ideal gases. Tetra Images - Jessica Peterson, Getty Images

This example problem demonstrates how to calculate the pressure of a gas system using the ideal gas law and the van der Waal's equation. It also demonstrates the difference between an ideal gas and a non-ideal gas.

## Van der Waals Equation Problem

Calculate the pressure exerted by 0.3000 mol of helium in a 0.2000 L container at -25 °C using
a. ideal gas law
b. van der Waals equation
What is the difference between the non-ideal and ideal gases?
Given:
aHe = 0.0341 atm·L2/mol2
bHe = 0.0237 L·mol

## How to Solve the Problem

Part 1: Ideal Gas Law
The ideal gas law is expressed by the formula:
PV = nRT
where
P = pressure
V = volume
n = number of moles of gas
R = ideal gas constant = 0.08206 L·atm/mol·K
T = absolute temperature
Find absolute temperature
T = °C + 273.15
T = -25 + 273.15
T = 248.15 K
Find the pressure
PV = nRT
P = nRT/V
P = (0.3000 mol)(0.08206 L·atm/mol·K)(248.15)/0.2000 L
Pideal = 30.55 atm
Part 2: Van der Waals Equation
Van der Waals equation is expressed by the formula
P + a(n/V)2 = nRT/(V-nb)
where
P = pressure
V = volume
n = number of moles of gas
a = attraction between individual gas particles
b = average volume of individual gas particles
R = ideal gas constant = 0.08206 L·atm/mol·K
T = absolute temperature
Solve for pressure
P = nRT/(V-nb) - a(n/V)2
To make the math easier to follow, the equation will be broken into two parts where
P = X - Y
where
X = nRT/(V-nb)
Y = a(n/V)2
X = P = nRT/(V-nb)
X = (0.3000 mol)(0.08206 L·atm/mol·K)(248.15)/[0.2000 L - (0.3000 mol)(0.0237 L/mol)]
X = 6.109 L·atm/(0.2000 L - .007 L)
X = 6.109 L·atm/0.19 L
X = 32.152 atm
Y = a(n/V)2
Y = 0.0341 atm·L2/mol2 x [0.3000 mol/0.2000 L]2
Y = 0.0341 atm·L2/mol2 x (1.5 mol/L)2
Y = 0.0341 atm·L2/mol2 x 2.25 mol2/L2
Y = 0.077 atm
Recombine to find pressure
P = X - Y
P = 32.152 atm - 0.077 atm
Pnon-ideal = 32.075 atm
Part 3 - Find the difference between ideal and non-ideal conditions
Pnon-ideal - Pideal = 32.152 atm - 30.55 atm
Pnon-ideal - Pideal = 1.602 atm