`H_(3)PO_(4)` is a tribasic acid with `pK_(a_(1)), pK_(a_(2))` and `pK_(a_(3)) 1.12, 7.21`, and `12.32`, respectively. It is used in fertiliser productions and its various salts are used in food, detrgent, toothpaste, and in metal treatment.
Small quantities of `H_(3)PO_(4)` are used in inparting the sour or tart taste of soft drinkes, such as Coca Cola, and beers, in which `H_(3)PO_(4)` is prese4nt `0.05%` by weight (density `= 1.0 gmL^(-1))`.
`10^(-3)M H_(3)PO_(4)(pH = 7)` is used in fertilisers as an aqueous soil digesting. Plants can absorb zinc in whater soluble from only. Zinc phosphate is the source of zinc and `PO_(4)^(3-`) ions in the soil. `K_(sp)` of zinc phosphate `= 9.1 xx 10^(-33)`.
Calculate the `pH` of a Coca Cola, assuming that the acidity of the cola arises only from `H_(3)PO_(4)` and `K_(a_(2))` and `K_(a_(3))` are no importance.

Options

A. `.18`
B. `2.2`
C. `3.3`
D. `4.4`

Explanation

To calculate the pH of Coca-Cola, we will follow these steps:

### Step 1: Calculate the concentration of H₃PO₄
Given that the weight percentage of H₃PO₄ in Coca-Cola is 0.05%, we can convert this to molarity.

1. **Weight percentage to molarity conversion**:
\[ \text{Concentration (M)} = \frac{\text{Weight \%} \times 1000 \, \text{g}}{\text{Molar mass of H}_3\text{PO}_4 \times 100} \]
The molar mass of H₃PO₄ (Phosphoric acid) is approximately 98 g/mol.

Substituting the values:
\[ \text{Concentration} = \frac{0.05 \times 1000}{98 \times 100} = \frac{50}{9800} \approx 5.1 \times 10^{-3} \, \text{M} \]

### Step 2: Use the formula for pH of a weak acid
Since we are assuming that only the first dissociation of H₃PO₄ is significant (Kₐ₂ and Kₐ₃ are negligible), we can use the following formula for the pH of a weak acid:
\[ \text{pH} = \frac{1}{2} \left( pK_{a1} - \log C \right) \]
Where:
- \( pK_{a1} = 1.12 \)
- \( C = 5.1 \times 10^{-3} \, \text{M} \)

### Step 3: Calculate pH
1. **Calculate \( \log C \)**:
\[ \log C = \log(5.1 \times 10^{-3}) \approx -2.29 \]

2. **Substituting into the pH formula**:
\[ \text{pH} = \frac{1}{2} \left( 1.12 - (-2.29) \right) = \frac{1}{2} \left( 1.12 + 2.29 \right) = \frac{1}{2} \times 3.41 \approx 1.705 \]

### Step 4: Final pH value
Thus, the calculated pH of Coca-Cola is approximately:
\[ \text{pH} \approx 1.71 \]