If I want the change in Gibbs's free energy for the reverse reaction, do I also take the reciprocal?

No, it's not. It's just the opposite sign.

Try not to confuse the properties of the equilibrium constant with a state function like #DeltaH#. Their common properties are:

#ul"Equilibrium Constants"#

For some equilibrium constant #K#, an altered equilibrium constant #K'# due to modifying a reaction would be as follows.

• Reversed reactions have #K' = K^(-1)#.
• Reactions with stoichiometric coefficients scaled by a constant #c# have #K' = K^c#, i.e. the whole thing is raised by that constant.

For some reaction

#nu_A A + nu_B B -> nu_C C + nu_D D#,

if you instead have

#2nu_C C + 2nu_D D -> 2nu_A A + 2nu_B B#,

what you end up with is #K' = (K^2)^(-1) = K^(-2) = 1/K^2#.

#ul"State Functions"#

For the change in some state function #DeltaY# (such as enthalpy, entropy, volume, temperature, pressure, etc), the altered change in the state function #DeltaY'# would be as follows.

• Reversed reactions have #DeltaY' = -DeltaY#.
• Reactions with stoichiometric coefficients scaled by a constant #c# have #DeltaY' = cDeltaY#, i.e. you just multiply by that constant.

For some reaction

#nu_A A + nu_B B -> nu_C C + nu_D D#,

if you instead have

#2nu_C C + 2nu_D D -> 2nu_A A + 2nu_B B#,

what you end up with is #DeltaY' = -2DeltaY#.