Question

If solution \(\mathrm{A}\) has a \(\mathrm{pH}\) that is three \(\mathrm{pH}\) units greater than that of solution B, how much greater is the \(\mathrm{H}_{3} \mathrm{O}^{+}\) concentration in solution B than in solution A?

Short answer

The concentration of \(\mathrm{H}_3 \mathrm{O}^+\) ions in solution B is 1000 times greater than in solution A.

Step-by-step solution

Identify given values and needed values

We know: \(\mathrm{pH(A)} - \mathrm{pH(B)} = 3\), and we want to find: \([\mathrm{H}_3 \mathrm{O}^+ _\mathrm{B}]\) and \([\mathrm{H}_3 \mathrm{O}^+ _\mathrm{A}]\)

Use the formula for pH

We use the formula: \(\mathrm{pH} = -\log [\mathrm{H}_3 \mathrm{O}^+]\). Therefore, \([- \log [\mathrm{H}_3 \mathrm{O}^+ _\mathrm{A}]] - [ - \log [\mathrm{H}_3 \mathrm{O}^+ _\mathrm{B}]] = 3\)

Simplify the expression

We simplify and add \(\log [\mathrm{H}_3 \mathrm{O}^+ _\mathrm{A}]\) to both sides: \(\log [\mathrm{H}_3 \mathrm{O}^+ _\mathrm{B}] - \log [\mathrm{H}_3 \mathrm{O}^+ _\mathrm{A}] = 3\)

Apply the logarithmic property

Applying the logarithm property \(\log a - \log b = \log (a/b)\), we obtain: \(\log \left( \frac{[\mathrm{H}_3 \mathrm{O}^+ _\mathrm{B}]}{[\mathrm{H}_3 \mathrm{O}^+ _\mathrm{A}]} \right) = 3\)

Solve for the ratio

To remove the logarithm, we must apply the antilogarithm (10 raised to the power of) to both sides: \(\frac{[\mathrm{H}_3 \mathrm{O}^+ _\mathrm{B}]}{[\mathrm{H}_3 \mathrm{O}^+ _\mathrm{A}]} = 10^3 = 1000\)

Key concepts

Understanding Hydronium Ion Concentration

Hydronium ion concentration, represented as \([\mathrm{H}_3 \mathrm{O}^+]\), is a key player in determining the acidity of a solution. The more hydronium ions, the more acidic the solution. This is directly linked to the concept of pH.
By definition, \(\mathrm{pH} = -\log [\mathrm{H}_3 \mathrm{O}^+]\). When the pH value changes, it indicates a change in the hydronium ion concentration.

• A lower pH means higher hydronium ion concentration.
• A higher pH means lower hydronium ion concentration.

In this exercise, Solution B has a hydronium ion concentration that is significantly greater than Solution A because it has a lower pH.
When the pH difference is 3, the hydronium ion concentration changes by a factor of 1000.This relationship highlights how even small pH changes can lead to large differences in ion concentration.

Exploring Logarithmic Properties

Logarithmic properties simplify complex calculations, especially in chemistry. The most relevant property for pH calculations is \(\log a - \log b = \log \left( \frac{a}{b} \right)\).
This property allows us to understand relative changes between two quantities, like hydronium ion concentrations.
In our exercise, \(\log [\mathrm{H}_3 \mathrm{O}^+ _\mathrm{B}] - \log [\mathrm{H}_3 \mathrm{O}^+ _\mathrm{A}] = 3\) translates to \(\log \left( \frac{[\mathrm{H}_3 \mathrm{O}^+ _\mathrm{B}]}{[\mathrm{H}_3 \mathrm{O}^+ _\mathrm{A}]} \right) = 3\).

• This tells us that the concentration ratio between solutions B and A is 1000.
• Applying the antilog (raising 10 to the power of 3) gives us the exact factor of difference.

Understanding these properties helps in performing pH and concentration calculations easily, making chemistry more approachable.

Basics of Acid-Base Chemistry

Acid-base chemistry focuses on the reaction of acids and bases in aqueous solutions. The concept of pH is central to this field.
When we talk about **acidic solutions**, we refer to those with high hydronium ion concentrations, usually with a pH less than 7.

• **Acids** increase the concentration of \(\mathrm{H}_3 \mathrm{O}^+\) ions.
• **Bases** increase the concentration of hydroxide ions but lower the \(\mathrm{H}_3 \mathrm{O}^+\) ions.