# 11.3.2 Equation of State for an Ideal Gas

It is impossible to track and analyse each and every atom or molecule in a system. Fortunately, a set of state variables, namely the pressure p, volume V and temperature T, can be used to describe the state of the system. An equation used to model the relationship among the state variables (of a given amount of substance) is called an equation of state.

The equation of state for an ideal gas is

$pV=nRT$

where n is the amount of substance in moles, and $R=8.31\text{ J }{{\text{K}}^{{-1}}}\text{ mo}{{\text{l}}^{{-1}}}$  is the molar gas constant.

Since each mole contains ${{N}_{A}}=6.02\times {{10}^{{23}}}$  particles (called the Avogadro’s number), the equation of state can also be written as

\begin{aligned}pV&=(n{{N}_{A}})(\frac{R}{{{{N}_{A}}}})T\\&=NkT\end{aligned}

where  N is the number of gas particles, and $\displaystyle k=\frac{R}{{{{N}_{A}}}}=1.38\times {{10}^{{-23}}}\text{ J }{{\text{K}}^{{-1}}}$  is the Boltzmann’s constant.

• When plugging numbers into this equation, one must ensure that p is the absolute pressure (not pressure relative to atmospheric pressure, aka gauge pressure), and T must be the thermodynamic temperature in Kelvin (not Celsius).
• Historically, the three empirical gas laws, namely

Boyle’s Law: ${{p}_{1}}{{V}_{1}}={{p}_{2}}{{V}_{2}}$  at constant T

Charles’ Law: $\displaystyle \frac{{{{V}_{1}}}}{{{{T}_{1}}}}=\frac{{{{V}_{2}}}}{{{{T}_{2}}}}$  at constant p

Gay-Lussac’s Law: $\displaystyle \frac{{{{p}_{1}}}}{{{{T}_{1}}}}=\frac{{{{p}_{2}}}}{{{{T}_{2}}}}$  at constant V

were first combined into what’s called the

Combined Gas Law: $\displaystyle \frac{{{{p}_{1}}{{V}_{1}}}}{{{{T}_{1}}}}=\frac{{{{p}_{2}}{{V}_{2}}}}{{{{T}_{2}}}}$

which is then combined with

Avogadro’s Law: $\displaystyle \frac{{{{V}_{1}}}}{{{{n}_{1}}}}=\frac{{{{V}_{2}}}}{{{{n}_{2}}}}$  at constant p and T

to finally arrive at

$pV=nRT$

Demonstration

Fountain Boy, Blue Danube, and Whistling Balloon

Concept Test

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