Calculate the #"pH"# of a solution that resulted when #"40 mL"# of a #"0.1-M"# ammonia solution was diluted to #"60 mL"# ?

Start by calculating the molarity of the diluted ammonia solution.

You know that your stock solution contains

#40 color(red)(cancel(color(black)("mL solution"))) * "0.1 moles NH"_3/(10^3color(red)(cancel(color(black)("mL solution")))) = "0.0040 moles NH"_3#

After you dilute this solution by adding enough water to increase its volume from #"40 mL"# to #"60 mL"#, its molarity will be--remember to use the volume of the solution in liters!

#["NH"_3] = "0.0040 moles"/(60 * 10^(-3) quad "L") = "0.0667 M"#

Now, ammonia will act as a weak base in aqueous solution.

#"NH"_ (3(aq)) + "H"_ 2"O"_ ((l)) rightleftharpoons "NH"_ (4(aq))^(+) + "OH"_ ((aq))^(-)#

By definition, the expression of the base dissociation constant, #K_b#, is given by

#K_b = (["NH"_4^(+)] * ["OH"^(-)])/(["NH"_3^(+)])#

As you know, an aqueous solution at #25^@"C"# has

#color(blue)(ul(color(black)(K_a * K_b = 1 * 10^(-14))))#

Notice that the problem gives you the acid dissociation constant, #K_a#, of the ammonium cation, ammonia's conjugate acid. This means that the base dissociation constant for ammonia is equal to

#K_b= (1 * 10^(-14))/(5.7 * 10^(-10)) = 1.75 * 10^(-5)#

Now, if you take #x# #"M"# to be the equilibrium concentration of the ammonium cations and of the hydroxide anions, you can say that, at equilibrium, the solution will also contain

#["NH"_3] = (0.0667 - x) quad "M"#

This basically means that in order for the reaction to produce #x# #"M"# of ammonium cations and #x# #"M"# of hydroxide anions, the concentration of ammonia must decrease by #x# #"M"#.

Plug this back into the expression of the base dissociation constant to find

#1.75 * 10^(-5) = (x * x)/(0.0667 - x)#

#1.75 * 10^(-5) = x^2/(0.0667 - x)#

The value of the base dissociation constant is small enough compared to the initial concentration of the acid to justify the approximation

#0.0667 - x ~~ 0.0667#

This means that you have

#1.75 * 10^(-5) = x^2/0.0667#

which gets you

#x = sqrt(0.0667 * 1.75 * 10^(-5)) = 1.08 * 10^(-3)#

Since #x# #"M"# represents the equilibrium concentration of the hydroxide anions, you can say that your solution contains

#["OH"^(-)] = 1.08 * 10^(-3) quad "M"#

Consequently, the #"pH"# of the solution, which can be found using the fact that at #25^@"C"#, an aqueous solution has

#color(blue)(ul(color(black)("pH + pOH = 14")))#

will be equal to

#"pH" = 14 - "pOH"#

Since

#color(blue)(ul(color(black)("pOH" = - log(["OH"^(-)]))))#

you can say that your solution has

#"pH" = 14 - [- log(1.08 * 10^(-3))] = color(darkgreen)(ul(color(black)(11.0)))#

The answer is rounded to one decimal place, the number of sig figs you have for your values.