Lemon juice normally has a `pH` of `2`. If all the acid the lemon juice is citric acid and there are no citrate salts present, then what will be the citric acid concentration `[Hcit]` in the lemon juice? (Assume that only the first hydrogen of citric acid is important)
`HCithArrH^(+)+Cit^(-)`, `K_(a)=8.4xx10^(-4) mol L^(-1)`

Options

A. `8.4 xx 10^(-4) M`
B. `4.2 xx 10^(-4) M`
C. `16.8 xx 10^(-4) M`
D. `12.0 xx 10^(-2) M`

Answer

To find the concentration of citric acid \([H_{cit}]\) in lemon juice with a pH of 2, we can follow these steps:

### Step 1: Understand the relationship between pH and \([H^+]\)
The pH of a solution is related to the concentration of hydrogen ions \([H^+]\) by the formula:
\[ \text{pH} = -\log[H^+] \]
Given that the pH of lemon juice is 2, we can calculate \([H^+]\):
\[ [H^+] = 10^{-\text{pH}} = 10^{-2} \, \text{mol L}^{-1} = 0.01 \, \text{mol L}^{-1} \]

### Step 2: Write the dissociation equation for citric acid
Citric acid \((H_{cit})\) dissociates in water as follows:
\[ H_{cit} \rightleftharpoons H^+ + Cit^- \]
For the first dissociation of citric acid, we can denote:
- Initial concentration of citric acid = \(C\)
- Change in concentration due to dissociation = \(x\)

### Step 3: Set up the equilibrium expression
At equilibrium, the concentrations will be:
- \([H_{cit}] = C - x\)
- \([H^+] = x\)
- \([Cit^-] = x\)

The equilibrium constant \(K_a\) for the dissociation of citric acid is given by:
\[ K_a = \frac{[H^+][Cit^-]}{[H_{cit}]} \]
Substituting the equilibrium concentrations, we have:
\[ K_a = \frac{x^2}{C - x} \]

### Step 4: Substitute known values
Given \(K_a = 8.4 \times 10^{-4} \, \text{mol L}^{-1}\) and \(x = [H^+] = 0.01 \, \text{mol L}^{-1}\):
\[ K_a = \frac{(0.01)^2}{C - 0.01} \]
This simplifies to:
\[ 8.4 \times 10^{-4} = \frac{0.0001}{C - 0.01} \]

### Step 5: Solve for \(C\)
Rearranging the equation to solve for \(C\):
\[ C - 0.01 = \frac{0.0001}{8.4 \times 10^{-4}} \]
Calculating the right side:
\[ C - 0.01 = \frac{0.0001}{0.00084} \approx 0.119 \]
Thus,
\[ C = 0.119 + 0.01 = 0.129 \, \text{mol L}^{-1} \]

### Step 6: Final concentration
To express this in standard form:
\[ C \approx 12.9 \times 10^{-2} \, \text{mol L}^{-1} \]

### Conclusion
The concentration of citric acid \([H_{cit}]\) in lemon juice is approximately:
\[ \boxed{0.129 \, \text{mol L}^{-1}} \]