Home / IB Mathematics SL 1.5 Laws of exponents AI HL Paper 1- Exam Style Questions

IB Mathematics SL 1.5 Laws of exponents AI HL Paper 1- Exam Style Questions

IB Mathematics SL 1.5 Laws of exponents AI HL Paper 1- Exam Style Questions- New Syllabus

Question

The pH scale is a measure of the acidity of a solution. Its value is given by the formula:

\( pH = -\log_{10}[H^+] \)

where \( [H^+] \) is the concentration of hydrogen ions in the solution (measured in moles per litre).

a) Calculate the pH value if the concentration of hydrogen ions is 0.0003.

The pH of milk is 6.6.

b) Calculate the concentration of hydrogen ions in milk.

The strength of an acid is measured by its concentration of hydrogen ions. A lemon has a pH value of 2 and a tomato has a pH value of 4.5.

c) Calculate how many times stronger the acid in a lemon is when compared to the acid in a tomato.

▶️ Answer/Explanation
Markscheme

a) To calculate the pH value for \( [H^+] = 0.0003 \):
Method 1 (Formula):
Use the pH formula:
\( pH = -\log_{10}(0.0003) \)

Calculate:
\( pH \approx 3.522878745 \)

Round to two decimal places:
\( pH \approx 3.52 \)

Method 2 (Calculator):
Compute using a calculator:
\( -\log_{10}(0.0003) \approx 3.522878745 \)
Round to two decimal places:
\( pH \approx 3.52 \)

Thus:
The pH value is \( 3.52 \).

b) To calculate the hydrogen ion concentration in milk (\( pH = 6.6 \)):
Method 1 (Formula):
Use the inverse logarithm:
\( [H^+] = 10^{-6.6} \)

Calculate:
\( [H^+] \approx 2.51188643151 \times 10^{-7} \)

Thus:
The concentration is \( 2.51 \times 10^{-7} \) moles per litre.

Method 2 (Calculator):
Compute using a calculator:
\( 10^{-6.6} \approx 2.51188643151 \times 10^{-7} \)

Thus:
The concentration is \( 2.51 \times 10^{-7} \) moles per litre.

c) To calculate how many times stronger the acid in a lemon (\( pH = 2 \)) is compared to a tomato (\( pH = 4.5 \)):
Method 1 (Formula):
Calculate hydrogen ion concentrations:
\( [H^+]_{\text{lemon}} = 10^{-2} = 0.01 \)
\( [H^+]_{\text{tomato}} = 10^{-4.5} \approx 0.0000316227 \)

Find the ratio:
\( \frac{[H^+]_{\text{lemon}}}{[H^+]_{\text{tomato}}} = \frac{10^{-2}}{10^{-4.5}} = 10^{4.5 – 2} = 10^{2.5} \)
\( 10^{2.5} \approx 316.227766017 \)

Round to three significant figures:
\( \approx 316 \)

Method 2 (Calculator):
Compute concentrations:
\( [H^+]_{\text{lemon}} = 0.01 \)
\( [H^+]_{\text{tomato}} \approx 0.0000316227 \)

Calculate ratio:
\( \frac{0.01}{0.0000316227} \approx 316.227766017 \)

Round to three significant figures:
\( \approx 316 \)

Thus:
The acid in a lemon is approximately \( 316 \) times stronger than the acid in a tomato.

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