Question

A certain acid, HA, has a vapor density of 5.11 g/L when in the gas phase at a temperature of 25°C and a pressure of 1.00 atm. When 1.50 g of this acid is dissolved in enough water to make 100.0 mL of solution, the pH is found to be 1.80. Calculate K_a for HA.

Short answer

K_a for the given acid is 1.74×10^-4.

Step-by-step solution

Calculating Concentration of Acid

As the vapor density is given, so molar mass can be calculated as:

$\begin{array}{l}\text{Molar\hspace{0.17em}Mass\hspace{0.17em}=\hspace{0.17em}2×Vapor\hspace{0.17em}density}\\ \text{Molar\hspace{0.17em}Mass\hspace{0.17em}=\hspace{0.17em}2×5.11}\\ \text{Molar\hspace{0.17em}Mass\hspace{0.17em}=\hspace{0.17em}10.22}\end{array}$

Concentration:

$\begin{array}{c}\text{Mole=}\frac{\text{Mass}}{\text{MolarMass}}\\ \text{Mole=}\frac{\text{1.50}}{\text{10.22}}\text{=0.147\hspace{0.17em}Mole}\\ \text{Concentration=}\frac{\text{Mole}}{\text{Volume(L)}}\text{=}\frac{\text{0.147}}{\text{0.1}}\text{=1.47\hspace{0.17em}M}\end{array}$

Calculating [H+]

H+ ion concentration can be calculated from pH

$\begin{array}{l}{\text{pH=-log[H}}^{\text{+}}\text{]}\\ {\text{[H}}^{\text{+}}{\text{]=10}}^{\text{-pH}}\end{array}$

For the given case,

${\text{[H}}^{\text{+}}{\text{]=10}}^{\text{-1.8}}\text{=0.016}$

Calculating Ka for Acid HA

For weak acids, [H⁺] ion concentration is calculates using K_a as

${\text{[H}}^{\text{+}}\text{]=}\sqrt{{\text{K}}_{\text{a}}\text{×C}}$

So, using this K_acan be calculated

$\begin{array}{c}\text{0.016=}\sqrt{{\text{K}}_{\text{a}}\text{×1.47}}\\ {\text{2.56×10}}^{\text{-4}}{\text{=K}}_{\text{a}}\text{×1.47}\\ {\text{K}}_{\text{a}}\text{=}\frac{{\text{2.56×10}}^{\text{-4}}}{\text{1.47}}{\text{=1.74×10}}^{\text{-4}}\end{array}$

Hence, K_a for the given acid is 1.74×10^-4.