# 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**_{A}**k**_{b}**T**

*= μ*

**PV**

**N**_{A}**k**_{b}**T**……….(4)

Here, μ = (**N/N _{A})** = the number of molecules of gas/ Avogadro number = Number of moles

Again, if ** N_{A}k_{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_{A}k_{b}** = Avogadro number x

**Boltzmann constant**…………….(6)

The value of R is thus can be easily calculated this way:

R =

**= Avogadro number x**

**N**_{A}**k**_{b}**Boltzmann constant**= (6.023×10

^{23}) 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}