The Ideal Gas Law — Algebra Under Pressure
The Ideal Gas Law
Every breath you take, every tire you inflate, every weather system on Earth obeys a single algebraic equation:
$$PV = nRT$$
where $P$ = pressure (Pa), $V$ = volume (m³), $n$ = moles of gas, $R = 8.314$ J/(mol·K) (universal gas constant), and $T$ = temperature (Kelvin).
This is a multivariate linear equation. If you know any three variables, you can solve for the fourth using basic algebra.
Boyle's Law (constant T and n)
$$P_1 V_1 = P_2 V_2$$
Double the pressure → halve the volume. This is inverse proportionality.
Charles's Law (constant P and n)
$$\frac{V_1}{T_1} = \frac{V_2}{T_2}$$
Heat a gas → it expands. A direct proportion.
A car tire contains 0.5 mol of air at 25°C (298 K) and has a volume of 12 liters. What is the pressure inside?
$$P = \frac{nRT}{V} = \frac{0.5 \times 8.314 \times 298}{0.012}$$
$$= \frac{1{,}238.8}{0.012} = 103{,}233 \text{ Pa} \approx 103 \text{ kPa}$$
That's about 1 atm — standard atmospheric pressure. The tire math is just multiplication, division, and unit conversion.
A diver descends to 20 m depth where pressure is 3 atm. Their lungs hold 6 L of air at the surface (1 atm). What volume would that air occupy at depth?
Using Boyle's Law:
$$V_2 = \frac{P_1 V_1}{P_2} = \frac{1 \times 6}{3} = 2 \text{ L}$$
Air compresses to one-third its volume. This is why divers need compressed air tanks — simple algebra explains the physics of diving.
$PV = nRT$ is one of the most used equations in science and engineering. It's pure algebra — no calculus needed. Proportions and division let you predict what happens to air in tires, lungs, weather balloons, and jet engines.