Video Transcript
Calculate the K𝑎 value of a
0.2-molar aqueous solution of propanoic acid with a concentration of H+ ions of 1.62
times 10 to the negative three molar. Give your answer to one decimal
place. (A) 4.8 times 10 to the negative
four moles per liter. (B) 6.6 times 10 to the negative
five moles per liter. (C) 1.6 times 10 to the negative
two moles per liter. (D) 5.2 times 10 to the negative
seven moles per liter. Or (E) 1.3 times 10 to the negative
five moles per liter.
Propanoic acid is CH3CH2COOH. We know this from the name stem
“prop,” meaning three carbons in the chain, “an” telling us there are single bonds
between carbon atoms, and the suffix -oic acid telling us this is a carboxylic acid
with a carboxyl group. Propanoic acid is a weak acid,
meaning in solution it dissociates to a small degree, forming an equilibrium
according to this equation. The dissociation products are the
ions CH3CH2COO− and the hydrogen ion. We are asked to find K𝑎, the acid
dissociation constant for this weak acid.
We are told the starting
concentration of the acid, 0.2 molar. And we are told the hydrogen ion
concentration in solution, 1.62 times 10 to the negative three molar. Let’s start by writing the
expression for K𝑎. K𝑎 is equal to the molar
concentrations of the ion products, which are the propanoate ion and the hydrogen
ion, multiplied with each other divided by the molar concentration of propanoic
acid, the reactant. Square brackets refer to molar
concentration, or molarity, which is moles per liter or moles per decimeter
cubed.
We are given the denominator value
in terms of molarity, 0.2 molar. And we are given the hydrogen ion
concentration in terms of molarity too, 1.62 times 10 to the negative three
molar. But we don’t know the concentration
of the propanoate ion. However, from the balanced
equation, we know that the molar ratio of the anion to the hydrogen cation is one as
to one. And so the concentration of the
propanoate anion must be the same as that of the hydrogen ion.
Let’s now put in our values. We get 1.62 times 10 to the
negative three molar for the concentration of the anions multiplied by 1.62 times 10
to the negative three molar for the concentration of the hydrogen ions divided by
0.2 molar, which is the concentration of propanoic acid. And the value of the answer is 1.31
times 10 to the negative five. These two units can cancel, and so
the unit in the answer is molar, which is moles per liter. Note that sometimes K𝑎 and its
relative K𝑏, the base dissociation constant, are expressed without units.
We were asked to give our answer to
one decimal place. So let’s round off, and we get 1.3
times 10 to the negative five moles per liter, which corresponds with answer option
(E). The K𝑎 value for this propanoic
acid equilibrium is 1.3 times 10 to the negative five moles per liter.