Question #1f142

An indicator is a weak acid or a weak base that has different colors in undissociated and dissociated states.

In your case, methyl orange is a weak acid that's red in solutions that have a pH lower than 3.1 and yellow in solutions that have a pH larger than 4.4.

Because hydrochloric acid is a strong acid, it will dissociate completely to form

#HCl_((aq)) rightleftharpoons H_((aq))^(+) + Cl_((aq))^(-)#

This means that the concentration of the #H^(+)# ions will be equal to that of the #HCl#

#[H^(+)] = "0.05 M"#

Now, the generic equilibrium reaction for methyl orange (or any weak acid indicator, for that matter) is

#HIn_((aq)) rightleftharpoons H_((aq))^(+) + In_((aq))^(-)#

The equilibrium constant is defined as

#K_a = ([H^(+)] * [In^(-)])/([HIn])#

Use the pKa value given for methyl orange to determine the value of #K_a#

#K_a = 10^(-pK_a) = 10^(-3.7) = 1.99 * 10^(-4) ~= 2.0 * 10^(-4)#

Since you know #[H^(+)]# and #K_a#, use them in the above equation

#K_a = ([H^(+)] * [In^(-)])/([HIn]) => ([In^(-)])/([HIn]) = K_a/([H^(+)])#

#([In^(-)])/([HIn]) = (2.0 * 10^(-4))/(0.05) = 4.0 * 10^(-3)#

This is how much of the methyl orange will be left in basic form (yellow)

#"%yellow" = ([In^(-)])/([HIn]) * 100 = 4.0 * 10^(-3) * 100 = color(green)("0.4%")#

This means that the acid form (red) will be

#"%red" = 100 - 0.4 = color(green)("99.6%")#

Because the pH of the solution is very low

#pH_("solution") = -log([H^(+)]) = -log(0.05) = 1.3#

almost all of the methyl orange will be in acidic form.