Give the thermodynamic derivation of van't Hoff reaction isotherm, and explain its significance?

The van't Hoff reaction isotherm is given by:

#((del barG)/(del xi))_(T,P) = DeltabarG^@(T) + RTlnQ#,

where:

• #((delbarG)/(del xi))_(T,P)# describes the infinitesimal change in the molar Gibbs' free energy as the reaction proceeds at constant temperature and pressure.
• #xi# is the extent of reaction in terms of #"mols"# (it's represented as #x# in ICE tables you see in general chemistry).
• #DeltabarG^@# is the standard change in molar Gibbs' free energy, the reference point, defined at standard pressure (#"1 bar"#), and is a function of only temperature.
• #RTlnQ# is the deviation from #DeltabarG^@# at the same temperature.

It allows you to find the deviation of the Gibbs' free energy away from equilibrium, or away from standard conditions, at the same temperature.

• Equilibrium is if #((del barG)/(del xi))_(T,P) = DeltaG = 0# and #Q = K#.
• Standard conditions is if #Q = 1#, i.e. if all the activities #a_i# are #1#, so that #((del barG)/(del xi))_(T,P) = DeltaG^@#.

An important distinction is that standard conditions has #Q = 1#, but equilibrium does not require #Q = 1#, since #K# does not have to be #1#. Equilibrium just needs #DeltaG = 0# and #Q = K#. Thus, the two ways to use this equation described above are indeed different.

To derive this, we begin from the definition of reaction progress based on the chemical potential #mu_i# (the molar Gibbs' free energy, #mu_i -= barG_i = G_i//n_i#) of substance #i# in the reaction

#overbrace(((delbarG)/(del xi))_(T,P))^"Reaction Progress" = sum_i nu_i mu_i#,#" "" "bb((1))#

where #nu_i# is the unitless stoichiometric coefficient of substance #i#, and is negative for reactants and positive for products.

The deviation of the chemical potential due to changes in activities #a_i# (nonideal concentrations) is given by:

#mu_i(T,P) = mu_i^@(T) + RTln a_i#,#" "" "bb((2))#

where #mu_i^@(T)# is the chemical potential defined at standard pressure (#"1 bar"#).

By substituting the right-hand side of #(2)# into the right-hand side of #(1)# for #mu_i# (we write #mu_i^@# from this point to mean it as a function of only temperature):

#((del barG)/(del xi))_(T,P) = sum_i nu_i (mu_i^@ + RTln a_i)#

#" "" "" "" " \ = sum_i (nu_imu_i^@ + RT nu_iln a_i)#

#" "" "" "" " \ = sum_i nu_imu_i^@ + RT sum_i nu_iln a_i#

Now, at standard pressure and the desired temperature,

#DeltabarG^@ = ul(sum_i nu_i mu_i^@)#.

You may have seen this in general chemistry as:

#DeltaG_(rxn)^@ = overbrace(sum_"products" nu_P DeltaG_(f,P)^@ - sum_"reactants" nu_R DeltaG_(f,R)^@)^"Gibbs' free energies of formation"#

Using the properties of logarithms,

#sum_i nu_i ln a_i = sum_i ln (a_i^(nu_i)) = ln (prod_i (a_i)^(nu_i))#.

The definition of the reaction quotient #Q# in terms of activities is

#Q = prod_i (a_i)^(nu_i) = (prod_"Products" (a_j)^(nu_j))/(prod_"Reactants" (a_i)^(nu_i))#

So, this really means that #ul(sum_i nu_i ln a_i = lnQ)#.

Therefore, we obtain the van't Hoff reaction isotherm:

#color(blue)(barul|stackrel(" ")(" "((del barG)/(del xi))_(T,P) = DeltabarG^@(T) + RTlnQ" ")|)#