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Problems · Problem 6.16

Q.The concentration of hydrogen ion in a sample of soft drink is 3.8 × 10⁻³M. what is its pH ?

Mahe DhseTextbookSubjective· 2mImportance★★★★★est
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✓ Free question

pH is the negative logarithm (base 10) of the hydrogen ion concentration. For [H+]=3.8×10−3 M[H^+] = 3.8 \times 10^{-3} \, \text{M}, the pH is approximately 2.42.

Why pH = –log[H⁺] works

pH is a convenient scale to express acidity without writing tiny numbers. The definition is:

pH=−log⁡10[H+]\text{pH} = -\log_{10} [H^+]

where [H+][H^+] is the molar concentration of hydrogen ions. The logarithm compresses a wide range (like 10−110^{-1} to 10−1410^{-14} M) into a 0–14 scale. A lower pH means higher [H+][H^+] — more acidic.

Here, [H+]=3.8×10−3[H^+] = 3.8 \times 10^{-3} M. That’s already in scientific notation, so we can directly apply the formula.

Step-by-step calculation

  1. Write the pH formula

pH=−log⁡10(3.8×10−3)\text{pH} = -\log_{10} (3.8 \times 10^{-3})

  1. Use the logarithm product rule log⁡(a×b)=log⁡a+log⁡b\log(a \times b) = \log a + \log b, so:

pH=−[log⁡10(3.8)+log⁡10(10−3)]\text{pH} = -[\log_{10}(3.8) + \log_{10}(10^{-3})]

  1. Simplify the powers of 10 log⁡10(10−3)=−3\log_{10}(10^{-3}) = -3, so:

pH=−[log⁡10(3.8)−3]=3−log⁡10(3.8)\text{pH} = -[\log_{10}(3.8) - 3] = 3 - \log_{10}(3.8)

  1. Find log⁡10(3.8)\log_{10}(3.8) You can use a calculator or recall that log⁡103.8≈0.5798\log_{10} 3.8 \approx 0.5798 (since 100.58≈3.810^{0.58} \approx 3.8).

log⁡10(3.8)≈0.5798\log_{10}(3.8) \approx 0.5798

  1. Complete the subtraction

pH=3−0.5798=2.4202\text{pH} = 3 - 0.5798 = 2.4202

Rounding to two decimal places (matching the given concentration’s precision):

pH≈2.42\text{pH} \approx 2.42

Watch out

A common mistake is forgetting the negative sign or misapplying the log rule. For example, writing pH=−log⁡(3.8×10−3)=−log⁡(3.8)−log⁡(10−3)\text{pH} = -\log(3.8 \times 10^{-3}) = -\log(3.8) - \log(10^{-3}) is correct, but then forgetting the minus sign in front of the whole expression gives a wrong positive value. Always keep the negative outside the brackets.

Tip

If you remember that log⁡(10−n)=−n\log(10^{-n}) = -n, the calculation simplifies to pH=n−log⁡(coefficient)\text{pH} = n - \log(\text{coefficient}). Here n=3n=3, so pH=3−log⁡(3.8)\text{pH} = 3 - \log(3.8) — a quick mental check.

✓Final answer

The pH of the soft drink is 2.42\boxed{2.42}.

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