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The Ideal Gas Law

The Ideal Gas Law is the equation of state for a hypothetical "ideal" gas. It relates the pressure, volume, temperature, and amount of a gas. While no real gas is truly ideal, this law is a very accurate approximation for most gases at high temperature and low pressure.

PV = nRT

Where:

  • P is the pressure (measured in Pascals, Pa).
  • V is the volume (measured in cubic meters, m³).
  • n is the amount of substance (measured in moles, mol).
  • R is the ideal gas constant (8.314 J/(mol·K)).
  • T is the absolute temperature (measured in Kelvin, K).

Assumptions of an Ideal Gas

For an ideal gas, we assume:

  1. The gas molecules occupy negligible space compared to the container volume.
  2. The molecules are in constant, random motion.
  3. There are no intermolecular forces between molecules.
  4. All collisions are perfectly elastic (no energy lost).

Unit Conversions

Common conversions for gas problems:
• Temperature: K = °C + 273.15
• Pressure: 1 atm = 101,325 Pa
• Volume: 1 L = 0.001 m³

The Combined Gas Law

When the amount of gas (n) is constant, we can relate the states of a gas before and after a change (like compression or heating):

P₁V₁ / T₁ = P₂V₂ / T₂

Deep Dive: Worked Examples

✅ Example 1: Standard Molar Volume

Calculate the volume occupied by 1.00 mole of an ideal gas at STP (Standard Temperature and Pressure: 0°C and 1.00 atm).

Given: n = 1.00 mol, P = 101,325 Pa, T = 273.15 K
Formula: V = nRT / P
Solution:
V = (1.00 × 8.314 × 273.15) / 101,325
V = 2270.97 / 101,325 = 0.02241 m³
V = 22.41 Liters

✅ Example 2: Pressure in a Cylinder

A 50.0 L cylinder contains 2.5 moles of Oxygen gas at 25°C. Calculate the pressure inside the cylinder.

Given: V = 0.050 m³, n = 2.5 mol, T = 25 + 273 = 298 K
Formula: P = nRT / V
Solution:
P = (2.5 × 8.314 × 298) / 0.050
P = 6193.93 / 0.050 = 123,878 Pa
P = 1.22 atm (or 123.9 kPa)

✅ Example 3: Finding Moles of Gas

A weather balloon has a volume of 500 m³ and is at a pressure of 30 kPa and a temperature of -50°C. How many moles of Helium are in the balloon?

Given: V = 500 m³, P = 30,000 Pa, T = -50 + 273 = 223 K
Formula: n = PV / RT
Solution:
n = (30,000 × 500) / (8.314 × 223)
n = 15,000,000 / 1854.0 = 8091 mol

✅ Example 4: Combined Gas Law (Compression)

An air bubble at the bottom of a lake (P₁ = 3 atm, T₁ = 10°C) has a volume of 2 cm³. It rises to the surface (P₂ = 1 atm, T₂ = 25°C). What is its new volume?

Given: P₁=3, V₁=2, T₁=283 K, P₂=1, T₂=298 K
Formula: V₂ = (P₁V₁T₂) / (P₂T₁)
Solution:
V₂ = (3 × 2 × 298) / (1 × 283)
V₂ = 1788 / 283 = 6.32 cm³

✅ Example 5: Number of Molecules

A vacuum chamber of 1 m³ is pumped down to 1 mPa at 20°C. How many molecules remain in the chamber?

Given: V = 1 m³, P = 0.001 Pa, T = 293 K
Find n: n = (0.001 × 1) / (8.314 × 293) = 4.105 × 10⁻⁷ mol
Molecules: N = n × NA (Avogadro's Number)
Solution:
N = (4.105 × 10⁻⁷) × (6.022 × 10²³) = 2.47 × 10¹⁷ molecules