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 kb is brought in, so that K = N kb, 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 kb

=> kb = PV / (NT) ……. (2)

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

PV= N kb T ………… (3)

PV= (N/NA) NAkb T
PV= μ NAkb T ……….(4)

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

Again, if NAkb = 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 = NAkb = Avogadro number x Boltzmann constant …………….(6)
The value of R is thus can be easily calculated this way:
R = NAkb = 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

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