Combined Gas Law or gas equation | the Ideal Gas Equation

We will derive the following here (1) combined gas law or gas equation & (2) the Ideal Gas equation. For these, we will use Boyle’s law, Charles’ law, Pressure law, and Avogadro’s law for gases.

The Combined Gas Law or Gas Equation

Boyle’s and Charles’ laws give separately variations of the volume of a gas with pressure and temperature respectively.

By combining these two laws, we get an equation which gives the simultaneous effect of the changes of pressure and temperature on the volume of the gas.

This is known as the combined gas law or gas equation.

According to Boyle’s law,
V ∝1/P (Constant T) …(i)

According to Charles’ law,

V ∝ T (Constant P) …(ii)

Combining equations (i), & (ii), the combined gas law can be written as :

V ∝ T/P

or PV ∝ T

or PV = kT (where k is a constant)

or PV / T = constant (k)

This is known as combined gas law or gas equation.

This gives, P1 V1 / T1 = P2 V2 / T2

Alternative Derivation for the Gas Equation | ideal gas equation

The gas equation can also be derived by combining Boyle’s and Charles’ laws as given below :

According to Boyle’s law, for a fixed mass of a gas at a constant temperature

V ∝1/P (Constant T and n) …(i)

According to Charles’ law,

V ∝ T (Constant P and n) …(ii)

According to Avogadro’s law

V ∝ n (Constant P and T) …(iii) (n = number of moles)

Combining equations (i), (ii), and (iii), the combined gas law can be written as :

V ∝ nT/P

or PV ∝ nT

or PV = nRT …………… (1)

where R is a constant of proportionality and is known as the gas constant.

The value of R is the same for all gases. Therefore, it is also called the universal gas constant.

The above equation (1) is known as the ideal gas equation.

The word ideal is used because there is no actual gas that completely obeys this equation.

However, the real gases behave ideally at high temperatures and low pressures.

Since the ideal gas equation expresses the quantitative relationship between the four variables that describe the state of a gas, it is also known as the equation of state for gases.