Why do we write the activity of solids and pure liquids in equilibrium constants at standard conditions as #1#?

Because the standard state of solids and liquids is their molar density at room temperature and pressure, which is equal to their "concentrations" anyway.

When dividing by the standard state within the equilibrium constant, one gets approximately #1#, since the activity coefficient approaches #1# as the activity approaches #1#. So, #a ~~ 1# for pure solids and liquids sufficiently close to #25^@ "C"# and #"1 bar".

Since solids and liquids are quite incompressible and their densities don't vary that much with temperature, "sufficiently close to #25^@ "C"# and #"1 bar"#" is rather flexible.

The standard equilibrium constant, or even the standard reaction quotient, is written to be unitless. Suppose we have this reaction:

#"AgCl"(s) rightleftharpoons "Ag"^(+)(aq) + "Cl"^(-)(aq)#

#K_(sp) = ((gamma_(Ag^(+))["Ag"^(+)]//c^@)(gamma_(Cl^(-))["Cl"^(-)]//c^@))/(gamma_(AgCl(s))["AgCl"(s)]//c_(AgCl(s))^@)#

where #c^@ = "1 M"# for aqueous solutions, and #c_(AgCl(s))^@ = barrho_(AgCl(s))#, the molar density of #"AgCl"(s)#.

We know that #"AgCl"(s)# is vastly insoluble in water. Something like #K_(sp) ~~ 1.8 xx 10^(-10)#. So, the standard state of #"AgCl"(s)# is its state at #25^@ "C"# and #"1 bar"#, which is obviously a solid.

The density of #"AgCl"# is #"5.56 g/cm"^3#. So, its molar density is:

#barrho = (5.56 cancel"g")/(cancel"1 mL") xx (1000 cancel"mL")/("1 L") xx ("1 mol AgCl")/(143.32 cancel"g AgCl")#

#=# #"38.79 mol/L"#

Now, to calculate its "concentration", we suppose we have #"1 L"# of it, and start from the mass density to calculate how many mols we have of it. That would give our "concentration", #["AgCl"(s)]#.

#cancel"1 L AgCl" xx (5.56 cancel"g")/(cancel"1 mL") xx (1000 cancel"mL")/(cancel"1 L") xx "1 mol AgCl"/(143.32 cancel"g AgCl")#

#=# #"38.79 mols"# contained in that #"1 L"#

Well, there you go, the concentration is equal to the molar density at the same temperature.

So, the activity of #"AgCl"(s)#, given by the activity coefficient #gamma#, concentration #[" "]#, and standard-state concentration #c^@# in #"mol/L"#, is:

#a_("AgCl"(s))^@ = gamma_("AgCl"(s)) cdot (["AgCl"(s)])/c^@ ~~ gamma_("AgCl"(s)) ~~ 1#

and #gamma -> 1# as #a -> 1#.

And so, the equilibrium constant could be written as:

#K_(sp) = ((gamma_(Ag^(+))["Ag"^(+)]//c^@)(gamma_(Cl^(-))["Cl"^(-)]//c^@))/(gamma_(AgCl(s))["AgCl"(s)]//c_(AgCl(s))^@)#

#~~ (gamma_(Ag^(+))["Ag"^(+)]//c^@)(gamma_(Cl^(-))["Cl"^(-)]//c^@)#

where #c^@ = "1 M"#, and #c_(AgCl(s))^@ = barrho_(AgCl(s)) = ["AgCl"(s)]#.

Particularly when the temperature is close to #25^@ "C"#, the approximation becomes sufficient.

At very nonstandard temperatures, eventually #a_("AgCl"(s))# is not sufficiently close to #1# anymore.