Lemon juice has a pH = 2.1 . If all the acid in lemon is citric acid `(H "Cit." hArr H^(+)+"Cit"^(-1))` and `K_(a)` for citric acid is `8.4xx10^(-4)` mole/litre, what is the concentration of citric acid in lemon juice ?

Explanation

To find the concentration of citric acid in lemon juice given its pH, we can follow these steps:

### Step 1: Calculate the concentration of H⁺ ions from pH
The pH of the lemon juice is given as 2.1. We can use the formula for pH to find the concentration of hydrogen ions (H⁺):

\[ \text{pH} = -\log[H^+] \]

To find [H⁺], we rearrange the formula:

\[ [H^+] = 10^{-\text{pH}} \]

Substituting the given pH:

\[ [H^+] = 10^{-2.1} \approx 7.94 \times 10^{-3} \, \text{mol/L} \]

### Step 2: Set up the expression for the dissociation of citric acid
Citric acid (H₃Cit) dissociates in the following manner:

\[ H_3Cit \rightleftharpoons H^+ + H_2Cit^- \]

The dissociation constant \(K_a\) for citric acid is given as \(8.4 \times 10^{-4}\). The expression for \(K_a\) is:

\[ K_a = \frac{[H^+][H_2Cit^-]}{[H_3Cit]} \]

### Step 3: Assume initial concentration of citric acid
Let the initial concentration of citric acid be \(C\) mol/L. At equilibrium, if \(x\) is the amount that dissociates, we have:

- \([H^+] = x\)
- \([H_2Cit^-] = x\)
- \([H_3Cit] = C - x\)

Since we know from Step 1 that \([H^+] = 7.94 \times 10^{-3}\), we can set \(x = 7.94 \times 10^{-3}\).

### Step 4: Substitute into the \(K_a\) expression
Now substituting into the \(K_a\) expression:

\[ K_a = \frac{(7.94 \times 10^{-3})(7.94 \times 10^{-3})}{C - 7.94 \times 10^{-3}} \]

This simplifies to:

\[ 8.4 \times 10^{-4} = \frac{(7.94 \times 10^{-3})^2}{C - 7.94 \times 10^{-3}} \]

### Step 5: Solve for C
Calculating \((7.94 \times 10^{-3})^2\):

\[ (7.94 \times 10^{-3})^2 \approx 6.2936 \times 10^{-5} \]

Now substituting back into the equation:

\[ 8.4 \times 10^{-4} = \frac{6.2936 \times 10^{-5}}{C - 7.94 \times 10^{-3}} \]

Cross-multiplying gives:

\[ 8.4 \times 10^{-4}(C - 7.94 \times 10^{-3}) = 6.2936 \times 10^{-5} \]

Expanding this:

\[ 8.4 \times 10^{-4}C - 6.67456 \times 10^{-6} = 6.2936 \times 10^{-5} \]

Adding \(6.67456 \times 10^{-6}\) to both sides:

\[ 8.4 \times 10^{-4}C = 6.2936 \times 10^{-5} + 6.67456 \times 10^{-6} \]

Calculating the right side:

\[ 8.4 \times 10^{-4}C = 6.961056 \times 10^{-5} \]

Now, divide both sides by \(8.4 \times 10^{-4}\):

\[ C = \frac{6.961056 \times 10^{-5}}{8.4 \times 10^{-4}} \approx 0.083 \, \text{mol/L} \]

### Conclusion
The concentration of citric acid in lemon juice is approximately \(0.083 \, \text{mol/L}\).

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