[ptcek]

Hi folks,

I'm learning physical chemistry and found a problem. I don't know how to derive fugacity as a function of temperature ...  f = f(T).

It should look like this at the final stage:

ln f(T,p) = ln f(T₀,p) - INT_T0^T {H_m - H_m^id}/{RT²} dT

where  f = fugacity
          T = thermodynamic temperature
          p = pressure
          Hm = molar enthalpy
          Hm^id = molar enthalpy of ideal gas
          R = gas constant

[Hunt]

Hello ptcek ,

You can use the gibbs-helmholtz equation for a system undergoing a change in temperature from T_o to T. Consider a real system versus an ideal system. Then you can write : 

d [ ( G - G_id ) / T ] / dT = - ( H - H_id ) / T² 

Then the classical relation G = G^o + RT Ln f ( f = P for an ideal system ) is substituted to yield :

d Ln ( f / p ) = - 1 / R [ ( H - H_id ) / T² ] dT

We can now integrate setting T between T_o and T on the right. On the left the Ln(f/p) changes as a function of T from T_o to  T. However P is the same for both states. Thus , Ln( 1/p ) cancels and you get :

Ln f ( T,p ) - Ln f ( T_o , p ) = - 1 / R Integral ( H - H_id ) / T² dT  with the integral limits already set. This is the relation required.