What is the ideal gas equation? Derive the equation for an ideal gas and describe the characteristics of $R$ and its value.

(N/A) Ideal gas equation: The equation obtained by combining Boyle's law,Charles's law,and Avogadro's law is called the "Ideal Gas Equation."
Ideal gas equation: $pV = nRT$ (Eq.-$i$)
The ideal gas equation is a relation between four variables $(p, V, n, T)$. It describes the state of any gas,so it is called the equation of state.
Derivation of the ideal gas equation:
$(i)$ Boyle's law: $V \propto \frac{1}{p}$ (at constant $T$ and $n$)
$(ii)$ Charles's law: $V \propto T$ (at constant $p$ and $n$)
$(iii)$ Avogadro's law: $V \propto n$ (at constant $p$ and $T$)
Combining these,$V \propto \frac{nT}{p}$
If the proportionality constant is $R$,then:
$V = R \left(\frac{nT}{p}\right)$
Rearranging gives: $pV = nRT$
Characteristics of the gas constant $R$:
- $R$ is known as the universal gas constant.
- The value of $R$ is the same for all gases.
- The value of $R$ depends on the units used for $p$,$V$,and $T$. For example,$R = 8.314 \ J \ K^{-1} \ mol^{-1}$ in $SI$ units.