You haven't given the relevant #sf(E^@)# values so I looked them up:
#sf(Au^(3+)+3erightleftharpoonsAucolor(white)(xxxxxxxx)E^@=+1.42color(white)(x)V)#
#sf(Au^(+)+erightleftharpoonsAucolor(white)(xxxxxxxxxx)E^(@)=+1.68color(white)(x)V)#
#sf(Au^(3+)+2erightleftharpoonsAu^(+)color(white)(xxxxxxx)##sf(E^@# is unknown
We need to convert these to free energy values and then apply the conservation of energy.
#sf(DeltaG^@=-nFE^@)#
#sf(F)# is the Faraday Constant
#sf(n)# is the no. of moles of electrons transferred.
Let #sf(E^@)# be the value we want to find.
We can say that:
#sf(DeltaG^@[Au^(3+)rarrAu]=DeltaG^@[Au^(3+)rarrAu^(+)]+DeltaG^@[Au^(+)rarrAu])#
#:.##sf(-3cancel(F)xx1.42=-2cancel(F)E^(@)-cancel(F)xx1.68)#
#:.##sf(-2E^@=-4.26+1.68=-2.58color(white)(x)V)#
#sf(E^@=+2.58/2=+1.29color(white)(x)V)#
