Question

A chemist makes four successive 10 -fold dilutions of \(1.0 \times 10^{-5} M \mathrm{HCl} .\) Calculate the \(\mathrm{pH}\) of the original solution and of each diluted solution (through \(1.0 \times 10^{-9} \mathrm{M} \mathrm{HCl}\) ).

Short answer

The \mathrm{pH} values are: Original 5, First 6, Second 7, Third 8, Fourth 9.

Step-by-step solution

Calculate the pH of the original solution

The original concentration of HCl is given as \(1.0 \times 10^{-5} M\). Since HCl is a strong acid, it dissociates completely in water. The concentration of \[H^+\] ions will be the same as the concentration of HCl.\[ [H^+] = 1.0 \times 10^{-5} M \] The pH is calculated using the formula: \[ \mathrm{pH} = -\log[H^+] \] Therefore, \[ \mathrm{pH} = -\log(1.0 \times 10^{-5}) = 5 \]

Calculate the pH of the first diluted solution

In the first dilution, the concentration of HCl is diluted 10-fold. The new concentration will be: \[ \text{New concentration} = 1.0 \times 10^{-5} M \div 10 = 1.0 \times 10^{-6} M \] The pH is calculated as follows: \[ \mathrm{pH} = -\log(1.0 \times 10^{-6}) = 6 \]

Calculate the pH of the second diluted solution

In the second dilution, the concentration of HCl is diluted another 10-fold. The new concentration will be: \[ \text{New concentration} = 1.0 \times 10^{-6} M \div 10 = 1.0 \times 10^{-7} M \] The pH is calculated as follows: \[ \mathrm{pH} = -\log(1.0 \times 10^{-7}) = 7 \]

Calculate the pH of the third diluted solution

In the third dilution, the concentration of HCl is diluted yet another 10-fold. The new concentration will be: \[ \text{New concentration} = 1.0 \times 10^{-7} M \div 10 = 1.0 \times 10^{-8} M \] The pH is calculated as follows: \[ \mathrm{pH} = -\log(1.0 \times 10^{-8}) = 8 \]

Calculate the pH of the fourth diluted solution

In the fourth dilution, the concentration of HCl is again diluted 10-fold. The new concentration will be: \[ \text{New concentration} = 1.0 \times 10^{-8} M \div 10 = 1.0 \times 10^{-9} M \] The pH is calculated as follows: \[ \mathrm{pH} = -\log(1.0 \times 10^{-9}) = 9 \]

Key concepts

Dilution

Dilution involves reducing the concentration of a solute in a solution, usually by adding more solvent. When you dilute a solution, the amount of solute stays the same, but its concentration decreases because the volume of the solution increases.
You can calculate the new concentration after dilution using this formula:
\[\text{New concentration} = \frac{\text{Original concentration}}{\text{Dilution factor}} \]
For example, if you start with a solution that has a concentration of 1.0 x 10^-5 M and dilute it 10-fold (which means you're increasing the volume by 10 times), the new concentration is calculated like this:
\[\text{New concentration} = \frac{1.0 \times 10^{-5} M}{10} = 1.0 \times 10^{-6} M \]

This process can be repeated multiple times. Each time, the concentration of HCl is reduced by 10, as shown in the steps in your problem. Every step involves dividing the previous concentration by 10.

Strong Acids

Strong acids like hydrochloric acid (HCl) completely dissociate in water. This means that when HCl is dissolved in water, it breaks apart into hydrogen ions (H+) and chloride ions (Cl-).
This complete dissociation is what makes strong acids very effective at increasing the concentration of hydrogen ions in a solution. For HCl:
HCl → H+ + Cl-

Because of this property, we can directly use the concentration of HCl to represent the concentration of H+ ions. For example, if you have a 1.0 x 10^-5 M solution of HCl, the concentration of H+ ions (which drives pH) is also 1.0 x 10^-5 M.
The strong acid property simplifies pH calculations significantly since you don't have to consider partial dissociation as you would with weak acids.

Logarithms

The pH of a solution is calculated using the negative logarithm of the hydrogen ion concentration. Logarithms are mathematical operations that help us handle very large or very small numbers more easily. The formula to find pH is:
\[\text{pH} = -\text{log}[H^+] \]

Let's break this down:

• The symbol [H+] stands for the concentration of hydrogen ions in the solution.
• The log function finds the exponent to which the base 10 must be raised to get that number.
• The negative sign flips the value from negative to positive since the hydrogen ion concentration is usually a very small number.

For example, if [H+] = 1.0 x 10^-5 M, the pH is:
\[\text{pH} = -\text{log}(1.0 \times 10^{-5}) = 5 \]

Understanding logarithms is crucial for pH calculations because it simplifies working with the small values common in chemistry.
You can always use a calculator with a log function to find these values, ensuring accuracy in your pH calculations.