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Lesson Directive // ThermodynamicsREF_CORE

The Ideal Gas Model

P

V

=

n

R

T

Hover over a variable in the formula above, or see glossary below:

P

Pressure

Pascals (Pa)

V

Volume

Cubic metres (m³)

n

Amount of Substance

Moles (mol)

R

Ideal Gas Constant

J/(mol·K)

T

Temperature

Kelvin (K)

An ideal gas is a theoretical model where gas molecules are perfectly elastic point particles that do not interact with each other. Real gases approximate this model well under low pressure and high temperature conditions.

INSIGHT: Pressure is caused by gas molecules bouncing off container walls millions of times per second.

The Four Laws Combined

PV = nRT combines Boyle's Law (P ∝ 1/V at constant T), Charles's Law (V ∝ T at constant P), and Avogadro's Law (V ∝ n) into a single unified equation for predicting gas behaviour.

INSIGHT: If you know any four of the five quantities, you can solve for the fifth.

Entropy & the Second Law

The Second Law of Thermodynamics states that the total entropy of an isolated system always increases. Heat flows spontaneously from hot to cold — never the reverse — because the disordered state is statistically overwhelmingly more probable.

INSIGHT: Entropy always increases. Ordered systems naturally tend toward disorder.

Detailed Theory & ReferencesEXT_DOC

The Ideal Gas Law and Thermodynamics

The Ideal Gas Law is the equation of state for a hypothetical ideal gas, a useful approximation for many real gases under standard conditions:

$PV = nRT$

State Variables

Symbol	Quantity	SI Unit
$P$	Pressure	Pascal (Pa)
$V$	Volume	Cubic metre (m³)
$n$	Amount of substance	Mole (mol)
$R$	Ideal gas constant	8.314 J/(mol·K)
$T$	Temperature	Kelvin (K)

The Four Laws it Unifies

Boyle's Law (1662): At constant temperature and amount, $PV = \text{const}$ , so $P \propto 1/V$ .
Charles's Law (1787): At constant pressure and amount, $V/T = \text{const}$ , so $V \propto T$ .
Gay-Lussac's Law (1809): At constant volume and amount, $P/T = \text{const}$ , so $P \propto T$ .
Avogadro's Law (1811): At constant pressure and temperature, $V/n = \text{const}$ , so $V \propto n$ .

The Laws of Thermodynamics

Zeroth Law: If system A is in thermal equilibrium with B, and B with C, then A is in equilibrium with C. (This defines temperature.)
First Law (Conservation of Energy): The change in internal energy equals heat added minus work done: $\Delta U = Q - W$ .
Second Law: The total entropy of an isolated system never decreases. Heat flows spontaneously from high to low temperature.
Third Law: As temperature approaches absolute zero, the entropy of a perfect crystal approaches a constant minimum.

Entropy

Entropy $S$ is a measure of the number of microscopic configurations $\Omega$ consistent with a macroscopic state: $S = k_B \ln \Omega$ where $k_B = 1.38 \times 10^{-23}$ J/K is the Boltzmann constant.

References

Thermal Physics (Blundell & Blundell, Oxford)
Feynman Lectures on Physics, Vol. I, Ch. 39–44
NIST: Thermophysical Properties

AI NOTICE

AI Assistance Disclaimer: This module uses AI-assisted educational models and interactive visual representations to help explain scientific and mathematical concepts. For formal research or academic evaluation, please verify formulas and data against standard primary reference materials.

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Syllabus Worked Example

Q: A sealed container holds n = 2 mol of gas at T = 300 K and V = 0.05 m³. Find the pressure P.

1.Use the ideal gas law: P = \frac{nRT}{V}.

2.Substitute: P = \frac{2 \times 8.314 \times 300}{0.05}.

3.Numerator: 2 \times 8.314 \times 300 = 4988.4 J.

4.Divide: P = \frac{4988.4}{0.05}.

Result:P \approx 99{,}768 \text{ Pa} \approx 0.985 \text{ atm}

The Ideal Gas Model

The Four Laws Combined

Entropy & the Second Law

¶The Ideal Gas Law and Thermodynamics

¶State Variables

¶The Four Laws it Unifies

¶The Laws of Thermodynamics