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I'm asked to find $\Delta G$ for the isothermal compression of $\ce{N2(g)}$ (real gas) at $0\ \mathrm{^\circ C}$ if the gas expands from $P=1\space\mathrm{atm}$ to $P=1000 \space\mathrm{atm}$. The fugacity coefficients are $\phi=0.9996$ and $\phi=1.84150$ respectively.

I'm confused about two things:

  1. My professor makes a distinction between $\phi^*$ and $\phi$, the first one being the fugacity coefficient for a pure gas and the second for a gas in a mixture. Are they the same in this case or did he make a mistake since there's not a mixture to begin with? (Not sure how Lewis—Randall rule can relate here since, again, there's no mixture to apply the relation between the two coefficients.)
  2. I already calculated the Gibbs free energy if the gas were an ideal one, I got $\Delta G=15.688\ \mathrm{kJ\over mol}$. Since $G^E=RT\ln(\gamma)$, and I have the Gibbs energy for an ideal gas, I could find the Gibbs energy for the real gas when I get a value for $G^E$; but again, in order to find $\gamma$ I need both coefficients, the one for a mixture and the one for a pure gas.

Can anyone please let me know what am I doing wrong? Or if my approach is far from the right path? Thanks in advance.

EDIT:

I assumed that the data that was given corresponded to the values of fugacity coefficients for the pure gas, making it all simpler since $G^E=RT\ln(\phi)$, so I got $G^E_{N_2}=17.075765\ \mathrm{kJ\over mol}$ and then $\Delta G^R=1.387765\ \mathrm{kJ\over mol}$

What I don't understand is that according to those two definitions of $G^E$, I get that $\gamma=\phi$, which I know it's not right, how's it possible?

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