The Ideal Gas Laws

As already mentioned, the behaviour of gases in terms of how volume, pressure and temperature are related can be described using the gas laws. We will now look at the gas laws in turn.

Boyle’s Law

The Irish scientist Sir Robert Boyle investigated the behaviour of gases when expanded or compressed under constant temperature (isothermal) conditions. In essence, Boyle’s Law states:

For any given mass of a gas, the absolute pressure will vary inversely with the volume providing that the temperature remains constant.

We can describe this mathematically as:

$p\alpha \frac{1}{V}$ OR $pV=\text{constant}$ 

e.g., Doubling the absolute pressure gives half the volume. Three times the absolute pressure gives one-third of the volume.

Boyle’s Law can also be expressed algebraically in the form:

p₁V₁ = p₂V₂ = p_nV_n for a specific mass of gas

Charles’s law

The French scientist Jacques Charles conducted experiments on gases where the pressure of a fixed mass of gas was kept constant while variations in the volume and temperature were examined. In essence, Charles’s Law states:

During the change of state of any gas in which the mass and pressure remain constant, the volume varies in proportion with the absolute temperature (Kelvin).

We can describe this mathematically as:

$V\alpha TOR\frac{V}{T}=\text{constant}$ 

e.g., At double the absolute temperature, the volume is doubled. At three times the absolute temperature, the volume is trebled.

Charles’s Law can be expressed algebraically in the form:

$\frac{V_1}{T_1}=\frac{V_2}{T_2}=\frac{V_n}{T_n}$ 

for a specific mass of gas.

Gay – Lussac’s Law (Constant Volume Process)

When a given mass of gas is heated at constant volume, its temperature and pressure will both increase. Conversely, if the gas is cooled, the temperature and pressure will both decrease. At any stage of either process, the ratio of the pressure p to the absolute temperature T of the gas will be a constant. Hence:

$\frac{\text{PRESSURE}}{\text{ABSOLUTE TEMPERATURE}}=\text{constant }OR\frac{p}{T}=C$ 

This is known as the Pressure Law or Gay – Lussac’s Law, which may be stated algebraically in the form:

$\frac{p_1}{T_1}=\frac{p_2}{T_2}=\frac{p_n}{T_n}$ 

for a specific mass and volume of gas.

The Combined Gas Law

If, during a process, the pressure, volume, and absolute temperature of a gas are changed from p1, V1, and T1 to p2, V2 and T2 respectively, then, provided there is no change in the mass of gas, Boyle’s Law, Charles’s Law, and the Pressure Law may be combined to give the algebraic expression:

$\frac{p_1{V}_1}{T_1}=\frac{p_2{V}_2}{T_2}=\text{constant}$ 

This is known as the Combined Gas Law.

The Characteristic Gas Equation

We have seen the combined gas law stated in the form:

$\frac{pV}{T}=\text{constant (C)}$ 

For a perfect or ideal gas, this constant C = mR where m is the mass of the gas and R is the Characteristic Gas Constant or Specific Gas Constant.

This means that:

$\frac{pV}{T}=mR$ 

Or

$pV=mRT$ 

This is known as the Characteristic Gas Equation of an ideal gas. When using this equation for solving problems, it is essential to express all the terms in appropriate units as shown in the table below:

The next table lists the gas laws, together with appropriate equations for problem solving.

The gas constant R, which appears in the Characteristic Gas Equation, is identified as the Characteristic Gas Constant or the Specific Gas Constant and its value varies for different gases as will be apparent in the questions covered in the self-assessments.

Universal Gas Constant

The Universal Gas Constant takes into account the concept of molecular mass of substances. This constant, symbol Ro, also known as the Molar Gas Constant, is the product of the relative molecular mass, M, and the Characteristic Gas Constant, R, and has the same value for all gases:

$R_0=MR=8.3143J{kg}^{-1}{k}^{-1}$ 

This means that the value of the Characteristic Gas Constant R can be found using:

$R=\frac{R_0}{M}$ 

e.g., The molecular mass of nitrogen is 28. What is the value of R for nitrogen?

$R=\frac{R_0}{M}=\frac{8.3143}{28}=0.297kJ{kg}^{-1}{k}^{-1}$ 

The universal gas constant is frequently utilised in problems dealing with the combustion of various gases and it appears in a version of the Characteristic Gas Equation called the Ideal Gas Equation. Our studies, however, will be restricted to the use of the Characteristic Gas Equation, which employs the Characteristic or Specific Gas Constant.