Question

How can you derive the ideal gas law?

Answer 1

Hint: The ideal gas law equation represents the application of combination of different gas laws which deals with ideal gasses. It can be used to generalize the behavior of gasses in different conditions. The ideal gas law equation shows the relation between pressure, volume, number of moles and temperature of a gas.

Complete answer:
The ideal gas law equation is given by
$PV=nRT$
Here, $P$ is the pressure of the gas
$V$ is its volume
$n$ is the number of moles of the gas
$R$ is the gas constant
$T$ is the temperature
Let us consider an ideal gas which is at pressure, $P$ and let its volume be $V$. The gas is kept at temperature, $T$ and the number of moles of the gas is $n$.
According to Boyle's law, at constant temperature and number of moles of the gas, the volume is inversely proportional to the pressure. Therefore,
$V\propto \dfrac{1}{P}$ -------- (1)
According to the Charles law, at constant pressure and number of moles of the gas, the volume is directly proportional to the temperature. Therefore,
$V\propto T$ -------- (2)
According to Avogadro's law, at constant pressure and temperature, the volume of the gas is directly proportional to the number of moles. Therefore,
$V\propto n$ -------- (3)
Combining eq (1), eq (2) and eq (3), we have,
$V\propto \dfrac{nT}{P}$
On removing the sign of proportionality,
$V=R\dfrac{nT}{P}$
Here,$R$ is the universal gas constant and is the constant of proportionality for the above equation.
Therefore, the ideal gas law is represented by the equation, $PV=nRT$.

Note:
Boyle's law, Charles’ law and Avogadro's law are known as ideal gas laws as they deal with ideal gases. The ideal gas equation is a good approximation of behavior of many gases under many different conditions. The value of universal gas constant is $8.314kJ\,mo{{l}^{-1}}{{K}^{-1}}$.