Pressure Exerted by an Ideal Gas: Full Derivation Made Easy

Pressure exerted by an ideal gas is the steady push per unit area on container walls caused by countless molecular collisions. The derivation shows that pressure depends on how many molecules are present, how massive they are, and how fast they move on average.

This derivation matters because it produces the famous result P = (1/3)ρc²rms and leads directly to the ideal gas equation PV = nRT. It is the bridge between molecular motion and the gas laws measured in laboratories, engines and the atmosphere.

Key Concepts

Consider N molecules each of mass m (kg) in a cubical box of side L, volume V = L3 (m3). A molecule has velocity components vx, vy, vz (m/s) with c2 = vx2 + vy2 + vz2. Momentum p = mv (kg·m/s). Pressure P = Force/Area (Pa = N/m2). Root mean square speed crms = √(mean of c2). Because motion is random, average vx2 = vy2 = vz2 = c2/3. Collisions with walls are elastic, so speed magnitude is unchanged, only the perpendicular component reverses.

Formulas

For one molecule hitting a wall perpendicular to x: momentum change = mvx − (−mvx) = 2mvx. Time between hits on the same wall = 2L/vx, so force from one molecule = (2mvx)/(2L/vx) = mvx2/L. Summing over all molecules, total force on that wall F = (m/L) Σvx2 = (Nm/L) × mean vx2. Dividing by area L2 gives P = (Nm/V) × mean vx2. Using mean vx2 = mean c2/3:

P = (1/3)(Nm/V) c²mean = (1/3) ρ crms2, where ρ = Nm/V is density (kg/m3). Also PV = (1/3) N m crms2 = (2/3) N × (½ m crms2), i.e. PV equals two-thirds of total translational kinetic energy. Assumptions: point molecules, elastic collisions, no intermolecular forces, large N, uniform distribution.

Worked Example

Calculate pressure of nitrogen at density ρ = 1.25 kg/m3 with crms = 500 m/s.

Step 1: P = (1/3) ρ crms2. Step 2: crms2 = 5002 = 250,000. Step 3: P = (1/3) × 1.25 × 250,000 = (1/3) × 312,500 ≈ 104,167 Pa. This is close to atmospheric pressure (101,325 Pa), showing the numbers are realistic for air near room temperature. If temperature rises, crms rises and so does pressure at fixed volume.

Common Misconceptions

• Misconception: Heavier molecules automatically exert more pressure at the same temperature. Correction: At fixed T, average kinetic energy ½mcrms2 is the same, so heavier molecules move slower; pressure depends on number density and temperature, not mass alone.
• Misconception: We can drop the 1/3 factor. Correction: The 1/3 comes from motion being shared among three dimensions; only one component pushes a given wall.
• Misconception: Pressure comes from molecules pushing each other. Correction: In the ideal model pressure comes only from wall collisions.

Real-World Applications

This result is used to estimate gas pressure in tyres, aerosol cans, gas pipelines, stellar interiors and vacuum chambers. It explains why heating a sealed spray can is dangerous (speed and pressure rise), how pressure gauges relate to molecular motion, and it is the starting point for deriving temperature, specific heats and transport properties of gases.

So pressure is not a mysterious fluid property — it is simply the averaged effect of molecular bombardment, quantified by P = (1/3)ρcrms2.