Question

A typical sample of vinegar has a pH of $3.0$. Assuming that vinegar is only an aqueous solution of acetic acid $(K_{\mathrm{a}}=1.8\times {10}^{-5})$, calculate the concentration of acetic acid in vinegar.

Short answer

The concentration of acetic acid in the vinegar is approximately 0.056 M.

Step-by-step solution

Understand the pH, Ka, and concentrations relationship

The pH is related to the concentration of hydrogen ions $[H^{+}]$ in the solution, and can be expressed as: $$pH=-{\log }_{10}[H^{+}]$$ The Ka (acid dissociation constant) is a measure of the strength of an acid in a solution and can be expressed as: $$K_a=\frac{[H^{+}][A^{-}]}{[HA]}$$ Where, - $[H^{+}]$ is the concentration of hydrogen ions - $[A^{-}]$ is the concentration of the conjugate base (acetate ions in this case) - $[HA]$ is the concentration of the acid (acetic acid in this case) Since vinegar is assumed to be an aqueous solution of acetic acid, we can presume that the $[A^{-}]$ present in the solution comes entirely from the dissociation of acetic acid and therefore, the concentration of acetate ions equals the concentration of hydrogen ions. Now we can write the Ka expression for this problem as: $$K_a=\frac{[H^{+}{]}^2}{[HA]-[H^{+}]}$$

Calculate the hydrogen ion concentration

Given the pH value of the vinegar is 3.0, we can calculate the hydrogen ion concentration using the pH formula: $$pH=-{\log }_{10}[H^{+}]$$ $$3.0=-{\log }_{10}[H^{+}]$$ To isolate $[H^{+}]$, we take the inverse logarithm of both sides: $$[H^{+}]={10}^{-3.0}$$ Now, calculate the numerical value of $[H^{+}]$: $$[H^{+}]=0.001\text{M}$$

Solve for the concentration of acetic acid

Use the Ka expression we derived earlier to solve for the concentration of acetic acid $[HA]$: $$K_a=\frac{[H^{+}{]}^2}{[HA]-[H^{+}]}$$ Plug in the values for Ka and the hydrogen ion concentration: $$1.8\times {10}^{-5}=\frac{(0.001{)}^2}{[HA]-0.001}$$ Rearrange the equation to solve for $[HA]$: $$[HA]-0.001=\frac{{0.001}^2}{1.8\times {10}^{-5}}$$ $$[HA]=0.001+\frac{{0.001}^2}{1.8\times {10}^{-5}}$$ Now, calculate the numerical value of $[HA]$: $$[HA]\approx 0.056\text{M}$$ This means that the concentration of acetic acid in the vinegar is approximately 0.056 M.

Key concepts

pH Calculation

The pH scale is a handy tool that helps us determine how acidic or basic a solution is. It ranges from 0 to 14, where 7 is neutral, below 7 is acidic, and above 7 is basic. The formula to calculate pH is:
$$pH=-{\log }_{10}[H^{+}]$$
This equation links pH with the concentration of hydrogen ions $[H^{+}]$ in a solution. In simple terms, the more hydrogen ions present, the lower the pH and the more acidic the solution is.
For example, if you have a pH of 3.0 in a vinegar solution, you can find the hydrogen ion concentration by rearranging the formula:

• $[H^{+}]={10}^{-3.0}$
• $[H^{+}]=0.001\text{M}$

Understanding this relationship is crucial because it connects pH directly with hydrogen ion concentration, allowing us to explore the acidity of various solutions.

Acetic Acid

Acetic acid is the main component of vinegar, which gives it that sour taste everyone recognizes. It's a weak acid, meaning it doesn't completely dissociate into ions in water. When acetic acid ($C{H}_{3}COOH$) dissolves in water, it partially breaks down into hydrogen ions ($[H^{+}]$) and acetate ions ($[C{H}_{3}CO{O}^{-}]$).
The strength of this dissociation is represented by the acid dissociation constant, $K_a$. For acetic acid, $K_a=1.8\times {10}^{-5}$. This value tells us how much of the acid dissociates in water.

• The larger the $K_a$, the stronger the acid.
• Acetic acid has a low $K_a$, indicating it's a weak acid.

In vinegar, we assume all the acetate ions come from acetic acid. This helps us set up our equations when calculating the acid's concentration in a solution.

Hydrogen Ion Concentration

Hydrogen ion concentration is a key player in determining the acidity of a solution. In an aqueous solution of acetic acid, the hydrogen ion concentration $[H^{+}]$ helps us understand how many hydrogen ions are present in the mixture, influencing the overall pH.
To calculate the concentration of acetic acid in vinegar, knowing $[H^{+}]$ is essential. Given the $pH=3.0$, we calculate that $[H^{+}]=0.001\text{M}$, showing how acidic the solution is.

• With $[H^{+}]$, we use the given $K_a$ to solve the equation:
• $$K_a=\frac{[H^{+}{]}^2}{[HA]-[H^{+}]}$$
• This helps determine the concentration of acetic acid $[HA]$, which is $0.056\text{M}$ for vinegar.

The interaction between $[H^{+}]$ and $K_a$ is fundamental in understanding the chemical behavior of acidic solutions like vinegar.