The Boltzmann Constant, k

We have two different formula for pV

 pV = nRT  

 pV = 1/3 Nmc²

We can equate them like this:

1/3 Nmc² = nRT

Now kinetic energy is 1/2 mv².  If we do a bit of rearranging:

   Nmc² = 3 nRT 

mc² = 3nRT
         N

1/2mc²  = 3nRT
                2N

Now for n moles of gas the total number of molecules N is equal to n x N_A where  N_A  is Avogadro's constant:     N = nN_A  substitute that into the above formula gives:

1/2mc²  = 3nRT  
                2nN_A

cancel the n which leads to:

1/2mc²  =   3    R    T  
              2   N_A

And this is a formula for kinetic energy

Where m = mass of one molecule
            c^{2 =} mean square speed of molecules in the gas (ms^-1)
            R = molar gas constant (= 8.3 J mol^-1 K^-1)
            T = absolute temperature (in Kelvin)
            N_A = Avogadro's constant (= 6.02 x 10^-23)

This is occasionally "simplified" ( ok ok don't laugh) by inventing yet another damn constant.  

If the   R   is replaced by  k   (the Boltzmann constant) then the formula reduces to 
           N_A

1/2mc²  = 3 kT  
          2

And yeah - all this needs to be learnt by heart

Boltzmann developed the kinetic theory  and used it to explain and predict many properties of gases.  Boltzmann's statistical methods have been used in many aspects of physics - including by Einstein and friends to describe electrons in atoms.