📂Thermal Physics

Ideal Gas Equation

Formulas¹

Let’s denote the number of molecules of a gas as $N$, volume as $V$, pressure as $p$, and absolute temperature as $T$. Then, the following equation holds, and this is called the ideal gas equation.

$$ pV = N k_{B} T $$

Here, $k_{B} = 1.3807 \times 10^{-23} J / K$ is called the Boltzmann constant.

Description

Historically, it was derived from experimental laws and later derived mathematically from kinetic theory of gases. It is called the ‘ideal gas’ equation because the following assumptions were made in the process of deriving the equation.

• There are no forces acting between each molecule. In other words, they do not attract each other.
• Each molecule is a point particle with no size.

In reality, molecules interact with each other and have size, but for the sake of simplicity of the theory, these assumptions are made. Just as Newtonian mechanics can explain many phenomena well without considering relativity on the surface of the earth, the ideal gas equation also well describes actual gases. An actual gas becomes closer to an ideal gas as the molecular weight of the system decreases, the temperature increases, and the pressure decreases.

The ideal gas equation does not explain all gas phenomena. When relativistic effects need to be considered, a relativistic gas model should be used, and when quantum effects need to be considered, a quantum gas model should be used.

The constant in the ideal gas equation can be expressed in the form of $pV = nRT$ with respect to the amount of substance $n$. Here, $R$ is called the gas constant, which is almost equally used in thermodynamics in the form.

Derivation

$$ p \propto \dfrac{1}{V} $$

At a constant temperature, the relationship between the pressure and volume of a gas is established, and this is called Boyle’s law. Later, independently of Boyle, Edme Mariotte also discovered the same fact, which is also called Boyle-Mariotte’s law.

$$ V \propto T $$

At a constant pressure, the relationship between the volume and temperature of a gas is established, and this is called Charles’ law.

$$ p \propto T $$

When the volume of a gas is constant, the relationship between temperature and pressure is established, and this is called Gay-Lussac’s law. From the three proportionalities above, the next equation is obtained.

$$ p^{2}V \propto T^{2}/V \implies p^{2}V^{2} \propto T^{2} \implies pV \propto T $$

If the proportionality constant is denoted as $Nk_{B}$, the following result is obtained.

$$ pV = Nk_{B}T $$