Ideal Gas vs Non-ideal Gas Example Problem

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.

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 Waal's equation

What is the difference between the non-ideal and ideal gases?

Given:

a_He = 0.0341 atm·L²/mol²
b_He = 0.0237 L·mol

Solution

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
P_ideal = 30.55 atm

Part 2: Van der Waal's Equation

Van der Waal's 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
P_non-ideal = 32.075 atm

Part 3 - Find the difference between ideal and non-ideal conditions

P_non-ideal - P_ideal = 32.152 atm - 30.55 atm
P_non-ideal - P_ideal = 1.602 atm

Answer:

The pressure for the ideal gas is 30.55 atm and the pressure for van der Waal's equation of the non-ideal gas was 32.152 atm. The non-ideal gas had a greater pressure by 1.602 atm.