understand that a gas obeying pV ∝ T, where T is the thermodynamic temperature, is known as an ideal gas

Equation of State

Learning Objective

Understand that a gas which obeys the relationship \$pV \propto T\$, where \$T\$ is the thermodynamic temperature, is called an ideal gas.

Key Concepts

• Pressure (\$p\$) – force exerted per unit area.
• Volume (\$V\$) – the space occupied by the gas.
• Thermodynamic temperature (\$T\$) – absolute temperature measured in kelvin (K).
• Proportionality – \$pV\$ is directly proportional to \$T\$ for a fixed amount of gas.

From Proportionality to the Ideal‑Gas Equation

If \$pV\$ is proportional to \$T\$, we can write

\$pV = k\,T\$

where \$k\$ is a constant for a given amount of gas.

When the amount of gas is expressed in moles (\$n\$), the constant becomes \$nR\$, where \$R\$ is the universal gas constant (\$R = 8.314\ \text{J mol}^{-1}\text{K}^{-1}\$). Thus the equation becomes

\$pV = nRT\$

This is the familiar ideal‑gas law.

Assumptions Behind the Ideal‑Gas Model

1. The gas particles are point masses with negligible volume.
2. There are no intermolecular forces except during elastic collisions.
3. Collisions with the container walls are perfectly elastic.
4. The gas is in thermal equilibrium, so a single temperature \$T\$ describes the kinetic energy of all particles.

Variables and Units

Sample Calculation

Calculate the pressure exerted by 2.00 mol of an ideal gas occupying a volume of 0.050 m³ at a temperature of 300 K.

Using \$pV = nRT\$:

\$p = \frac{nRT}{V} = \frac{(2.00\ \text{mol})(8.314\ \text{J mol}^{-1}\text{K}^{-1})(300\ \text{K})}{0.050\ \text{m}^3}\$

\$p = \frac{4988.4\ \text{J}}{0.050\ \text{m}^3} = 9.98\times10^{4}\ \text{Pa}\$

Thus the pressure is approximately \$1.0\times10^{5}\ \text{Pa}\$ (or 1 atm).

Limitations of the Ideal‑Gas Approximation

• At very high pressures, the volume of particles is no longer negligible.
• At low temperatures, intermolecular attractions become significant.
• Real gases deviate from \$pV = nRT\$; corrections are given by equations such as the \cdot an der Waals equation.

Summary

A gas that satisfies the proportionality \$pV \propto T\$ for a fixed amount of substance follows the ideal‑gas law \$pV = nRT\$. This simple relationship is a cornerstone of thermodynamics and provides a useful first approximation for the behaviour of many gases under ordinary conditions.