A weak acid,HA has the concentration of C mol/dm and degree of dissociation ,β.Assuming Ka<<1, prove that β is inversely proportional to square root of C?
Here's how you can prove this.
Before doing anything else, make sure that you understand what degree of dissociation means.
As you know, weak acids and weak bases do not dissociate completely in aqueous solution to form ions.
This means that at a given temperature, the number of molecules of weak acid that will dissociate to produce hydronium ions,
Simply put, not all molecules of acid will dissociate; some will do, but most will not. This is what degree of dissociation tells you, i.e. what Fraction of the acid molecules dissociate.
In your case, the degree of dissociation for the weak acid
For simplicity, I'll use hydrogen ions,
The dissociation of a weak acid looks like this
#"HA"_text((aq]) rightleftharpoons "H"_text((aq])^(+) + "A"_text((aq])^(-)#
As you know, the acid dissociation constant,
#K_a = ( ["H"^(+)] * ["A"^(-)])/(["HA"])#
You know that
#["HA"] = c#
Since we've established that the degree of dissociation tells you what fraction of the acid molecules dissociate, you can say that
#["H"^(+)] = beta * c" "#and #" "["A"^(-)] = beta * c#
Plug this into the equation for
#K_a = ( beta * color(red)(cancel(color(black)(c))) * beta* c)/color(red)(cancel(color(black)(c))) = beta^2 * c#
Your goal now is to isolate
#beta^2 = K_a * 1/c#
Take the square root of both sides to get
#beta = overbrace(sqrt(K_a))^(color(blue)("constant")) * 1/sqrt(c)#
#color(green)(beta prop 1/sqrt(c))#
Indeed, the degree of dissociation is inversely proportional to the square root of
#c uarr implies 1/sqrt(c) darr" "#and #" " c darr implies 1/sqrt(c) uarr#