Ideal Gas Equation by Academic Torch

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Ideal Gas Equation — Gas Laws (15.2.2)

15.2.2 — Ideal Gas Equation

Physics • Chapter 15 — Molecular Theory of Gases

The Ideal Gas Equation unifies the basic gas laws into one mathematical relation that describes the state of an ideal gas. Below is a step-by-step derivation showing how Boyle’s, Charles’s and Avogadro’s laws combine to give PV = nRT.

Ideal Gas Equation is given as:

PV = nRT

where:
P = pressure, V = volume, n = moles, R = gas constant, T = temperature (K)

Derivation — step by step

We start from the three empirical gas laws (each holds when the other variables are held constant) and combine them.

1) Boyle’s law (at constant T and n):
V ∝ 1 / P
or equivalently P V = k_B (constant for a given T and n)

2) Charles’s law (at constant P and n):
V ∝ T
or equivalently V / T = k_C (constant for a given P and n)

3) Avogadro’s law (at constant P and T):
V ∝ n
or equivalently V / n = k_A (constant for a given P and T)

Because volume is proportional to each of these variables (when others are fixed), we can combine the proportionalities into a single relation that captures dependence on all three variables:

From (1), (2), and (3):
V ∝ (n × T) / P
This is read as: volume increases with the number of moles and temperature, and decreases with pressure.

Remove the proportionality by introducing a constant of proportionality k (which depends on the system of units):

V = k ⋅ (n T) / P
Multiply both sides by P:
P V = k ⋅ n T
Rename the constant k as the universal gas constant R (this is a convention that makes the constant independent of the particular gas when expressed in appropriate units):
boxed:   P V = n R T
DO YOU KNOW?

The gas constant R has several common values depending on units: 8.314 J·mol⁻¹·K⁻¹ (SI) or 0.082057 L·atm·mol⁻¹·K⁻¹ (when using litres and atmospheres).

Graphical Representation

The Ideal Gas Equation relates three (or four) variables. At fixed n and T, PV = constant (Boyle); at fixed P and n, V ∝ T (Charles); at fixed P and T, V ∝ n (Avogadro).

Figure 15.7 — PV = nRT Isotherm
At constant T: isotherm PV = constant (Boyle). At constant P or V, curves resemble Charles or Avogadro behaviour.

Key Takeaways

  • PV = nRT — derived by combining Boyle’s, Charles’s, and Avogadro’s laws.
  • The relation V ∝ (n T)/P captures the combined dependence before introducing R.
  • Choosing the constant R gives a universal constant useful across gases (with appropriate units).
  • Use consistent units: P in Pa (or atm), V in m³ (or L), n in mol, T in K.
Read Avogadro’s Law →

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