Thermodynamics · Chapter 11

Ideal Gas Law

PV = nRT connects pressure, volume, temperature, and amount of gas into a single elegant relation — derived from nothing more than counting molecular collisions.

Prerequisites

Heat and Temperature \cdot Kinetic theory basics

Learning Goals

$State Boyle's, Charles's, Gay-Lussac's, and Avogadro's laws as special cases of PV = nRT$ .
Apply the ideal gas law to find P, V, n, or T given the other three quantities.
$Derive PV = Nk_BT from Newton's laws applied to molecular collisions$ .
Calculate internal energy and molar heat capacity using the equipartition theorem.
Identify when real gases deviate from ideal behavior and apply the van der Waals correction.

11.1 The Gas Laws

Three empirical gas laws, each discovered independently in the 17th–19th centuries, all turn out to be special cases of a single unified law:

Definition 11.1 — The Empirical Gas Laws

Boyle's Law ( $const T): PV = constant \qquad \to \qquad P_{1}V_{1} = P_{2}V_{2}$
Charles's Law ( $const P): V/T = constant \qquad \to \qquad V_{1}/T_{1} = V_{2}/T_{2}$
Gay-Lussac's Law ( $const V): P/T = constant \qquad \to \qquad P_{1}/T_{1} = P_{2}/T_{2}$
Avogadro's Law ( $const T, P): V \propto n$

Boyle noticed that halving the volume of a gas doubles its pressure — the molecules hit the walls twice as often. Charles observed that heating a gas at constant pressure makes it expand proportionally to absolute temperature. These combine into:

11.2 The Ideal Gas Law

PV=nRT

(11.1)

Here P is pressure (Pa), V is volume (m³), n is the amount of gas (moles), R = 8.314 J/mol·K is the universal gas constant, and T is absolute temperature (Kelvin). An equivalent form uses the number of molecules N and Boltzmann's constant k_B = R/N_A:

PV=Nk_BT \qquad k_B=1.381\times10^{-23}\,\mathrm{J/K}

(11.2)

An ideal gas is one in which (1) molecular volume is negligible compared to container volume, (2) molecules interact only via brief elastic collisions, and (3) there are no intermolecular attractive forces. Real gases obey this law closely at low pressure and high temperature.

Example 11.1 — Bicycle Tire Pressure

A bicycle tire has volume 1.2 L and is filled to gauge pressure 6. $0 atm at 20°C. The tire heats to 40°C in the sun. Find the new pressure. (Gauge pressure =$

Setup:

Volume is constant, so use Gay-Lussac: P_{1}/T_{1} = P_{2}/T_{2}

Absolute pressures:

P_{1} = 6.0 + 1.0 = 7.0 atm, T_{1} = 293 K, T_{2} = 313 K

New pressure:

P_{2} = P_{1}(T_{2}/T_{1}) = 7.0 \times (313/293) =

7.48 atm

absolute = 6.48 atm gauge

11.3 Kinetic Theory Derivation

The ideal gas law is not just empirical — it can be derived from Newton's laws applied to point-mass molecules. Consider N molecules in a cubic box of side L. Each molecule bouncing off a wall delivers impulse 2mv_x. The average force on one wall:

F=\frac{Nm\langle v_x^2\rangle}{L} \qquad P=\frac{F}{L^2}=\frac{Nm\langle v_x^2\rangle}{V}

(11.3)

Using isotropy (v²_x = v²_y = v²_z = v²_rms/3) and the definition of temperature (½mv²_rms = (3/2)k_BT), this gives PV = Nk_BT exactly.

Theorem 11.1 — Internal Energy of an Ideal Gas

The total internal energy of a monatomic ideal gas (3 translational degrees of freedom) is:

U = (3/2)Nk_BT = (3/2)nRT

For a diatomic gas (5 degrees of freedom — 3 translational, 2 rotational): U = (5/2)nRT. T

he molar heat capacity at constant volume is C_{V} = (f/2)R where f is the number of degrees

of freedom.

Figure 11.1. Kinetic gas simulation. Observe Boyle's law: compress the volume (slider) while watching pressure rise. Observe Ch

arles's law: raise temperature while watching the gas expand at constant pressure. Color =

11.4 Real Gases and the van der Waals Equation

At high pressure or low temperature, the ideal gas law fails because molecular volume and intermolecular attraction become significant. The van der Waals equation corrects for these:

\left(P+a\frac{n^2}{V^2}\right)(V-nb)=nRT

(11.4)

The term an²/V² accounts for attractive forces reducing effective pressure; nb accounts for the excluded volume of the molecules. The constants a and b are different for every gas. For CO₂: a = 3.64 L²·atm/mol², b = 0.0427 L/mol.

Example 11.2 — Volume of One Mole of Gas

$What volume does 1 mol of ideal gas occupy at STP (T = 273.15 K, P = 101.325 kPa$ ?

Ideal gas law:

V = nRT/P = (1)(8.314)(273.15)/(101325) = 2271/101325

Result:

V =

$0.02241 m^{3} = 22.41 L$ — the molar volume at STP.

Note:Every gas has the same molar volume at STP — this is Avogadro's principle.

Definition 11.2 — Common Traps

Gauge pressure is not absolute pressure: $add atmospheric pressure before using PV = nRT$
Gas laws require kelvin: proportionality to temperature fails if Celsius is used.
Units must match R: $use pascals and cubic meters with 8.314 J/(mol\cdotK), or use a matching atm\cdotL value$
Ideal behavior is an approximation: high pressure and low temperature require real-gas corrections.

Exercises — 11.1–11.4 Ideal Gas Law

A gas at 1.0 atm occupies 4.0 L. It is compressed to 2.0 L at constant temperature. What is the new pressure?

atm

Straightforward

A gas at 273 K occupies 2.0 L. At constant pressure its volume doubles. Find the new temperature in Kelvin.

Straightforward

Find the volume of 0.5 mol of ideal gas at T = 400 K and P = 1 atm. (R = 8.314 J/mol\cdotK, 1

atm = 101325 Pa

m³

Intermediate

How many molecules are in 1.0 L of ideal gas at 1.0 atm and 20°C? (k_{B} = 1.381\times10^{-23} J/K

molecules

Intermediate

5.A gas balloon contains 10 L of gas at 1.0 atm and 300 K. It is compressed to 3.0 atm and heated to 450 K. Find the new volume.

Challenging

Key Takeaways

$PV = nRT unifies Boyle's, Charles's, Gay-Lussac's, and Avogadro's laws$ .
Always use Kelvin in gas law calculations — never Celsius.
$Ideal gas internal energy: U = (f/2)nRT, where f = degrees of freedom (3 monatomic, 5 diat$ omic).
One mole of ideal gas at STP occupies 22.4 L regardless of which gas.
Real gases depart from ideal behavior at high pressure and low temperature — use van der Waals.