What is the pH of 0.1 M ammonia solution if Ka for the ammonium ion is 5.62 x 10-10 and Kw = 10-14?

##p"H" = 11.13##

The idea here is to use the ammonium ion's acid dissociation constant, ##K_a##, to find the value of ammonia's base dissociation constant, ##K_b##.

The relationship between the base dissocaition constant, the acid dissociation constant, and water's self-ionization constant, ##K_W##, is given by the equation

##K_W = K_a xx K_b##

In your case, the base dissociation constant of ammonia will be

##K_b = K_W/K_a = 10^(-14)/(5.62 * 10^(-10)) = 1.78 * 10^(-5)##

To determine the of the solution, you first need to know the ##p"OH"## of the solution, which in turn requires the cocnentration of hydroxide ions, ##"OH"^(-)##.

Ammonia will react with water to form ammonium and hydroxide ions. Use an ICE table to determine the equilibrium concentration of hydroxide ions

##"NH"_text(3(aq]) " "+" " "H"_2"O"_text((l]) " "rightleftharpoons" " "NH"_text(4(aq])^(+) " "+" " "OH"_text((aq])^(-)##

##color(purple)("I")" " " " 0.1" " " " " " " " " " " " " " " " " " " "0" " " " " " " " " " " "0## ##color(purple)("C")" " color(white)(x)-x" " " " " " " " " " " " " " " " " "(+x)" " " " " " " " "(+x)## ##color(purple)("E")" " color(white)(x)0.1-x" " " " " " " " " " " " " " " "color(white)(xx)x" " " " " " " " " "color(white)(x)x##

By definition, the base dissociation constant will be

##K_b = (["NH"_4^(+)] * ["OH"^(-)])/(["NH"_3]) = (x * x)/(0.1 - x)##

Because the value of ##K_b## is so small, you can say that

##(0.1 - x) ~~ 0.1##

This means that you have

##K_b = x^2/0.1 = 1.78 * 10^(-5)##

##x = sqrt(0.1 * 1.78 * 10^(-5)) = 0.001334##

This means that you have

##["OH"^(-)] = x = "0.001334 M"##

The ##p"OH"## of the solution will be

##p"OH" = -log(["OH"^(-)])##

##p"OH" = -log(0.001334) = 2.87##

The pH of the solution will thus be

##p"H" = 14 - p"OH" = 14 - 2.87 = color(green)(11.13)##