Question

Derive the ideal gas equation.

Answer 1

The ideal gas equation is an equation, which relates all the properties of an ideal gas. An ideal gas obeys Boyle’s law and Charles’ law and Avogadro’s law.
According to Boyle’s law,
PV = constant ……. (1)
According to Charles’s law,
`"V"/"T"`= constant ……. (2)
According to Avogadro’s law,
`"v"/"n"`= constant ……. (3)
After combining equations (1), (2), and (3), you can get the following equation.
`("p""v")/(n"T"`= constant ……. (4)
The above relation is called the combined law of gases. If you consider a gas, which contains μ moles of the gas, the number of atoms contained will be equal to p times the Avogadro number, NA
i.e. n = μN_A ……. (5)
Using equation (5), equation (4) can be written as
PV/ μN_AT = constant
The value of the constant in the above equation is taken to be k_B, which is called as Boltzmann constant (1.38 × 10^-23 JK^-1).
Hence, we have the following equation:
PV / μN_AT = k_B
PV = μN_Ak_BT
Here, μN_Ak_B = R, which is termed as universal gas constant whose value is 8.31 J mol^-1
K^-1.

PV = RT ……. (6)
The ideal gas equation is also called an equation of state because it gives the relation between the state variables and it is used to describe the state of any gas.