How to derive the value of R (Universal Gas Constant)

The Combined Gas Law or Gas Equation is as follows: PV / T = constant (K) ……… (1)

The value of this constant K changes if the volume of the gas changes. (so, not a constant actually)

Hence, a new constant k_b is brought in, so that K = N k_b, where n = number of molecules of gas.

Now, if we rewrite equation (1) with this new constant, then it becomes like this:

PV / T = N k_b

=> k_b = PV / (NT) ……. (2)

This constant k_b is known as the Boltzmann constant, and its value is 1.38 x 10^-23 JK^-1

PV= N k_b T ………… (3)

PV= (N/N_A) N_Ak_b T
PV= μ N_Ak_b T ……….(4)

Here, μ = (N/N_A) = the number of molecules of gas/ Avogadro number = Number of moles

Again, if N_Ak_b = Avogadro number x Boltzmann constant. This is also a constant. And this constant is designated as R and named as Universal Gas Constant.

Thus, equation (4) now can be written as PV= μ R T ……….. (5)

So, Universal Gas Constant = R = N_Ak_b = Avogadro number x Boltzmann constant …………….(6)
The value of R is thus can be easily calculated this way:
R = N_Ak_b = Avogadro number x Boltzmann constant = (6.023×10²³ ) x (1.38 x 10^-23) = 8.31 Joule mole^-1 K^-1

Thus we can derive the value of the Universal gas constant R as 8.31 Joule mole^-1 K^-1