The acid dissociation constant of uric acid is `K_(a)=4.0xx10^(-6)` M. The pH of a sample of urine is 6.0. What is the ratio of concentration of urate ion to uric acid in the urine?
(a)`2.0`
(b)`4.0`
(c)`6.0`
(d)`0.25`

Options

A. `2.0`
B. `4.0`
C. `6.0`
D. `0.25`

Answer

To solve the problem of finding the ratio of the concentration of urate ion to uric acid in urine, we can follow these steps:

### Step 1: Understand the relationship between pH, pKa, and the concentrations of the acid and its conjugate base.
The relationship can be described by the Henderson-Hasselbalch equation:
\[ \text{pH} = \text{pKa} + \log\left(\frac{[\text{A}^-]}{[\text{HA}]}\right) \]
where:
- \([\text{A}^-]\) is the concentration of the conjugate base (urate ion),
- \([\text{HA}]\) is the concentration of the acid (uric acid).

### Step 2: Calculate pKa from Ka.
Given:
\[ K_a = 4.0 \times 10^{-6} \, \text{M} \]
To find pKa, we use the formula:
\[ \text{pKa} = -\log(K_a) \]
Calculating this:
\[ \text{pKa} = -\log(4.0 \times 10^{-6}) = -(\log(4.0) + \log(10^{-6})) = -(\log(4.0) - 6) \]
Using a calculator, \(\log(4.0) \approx 0.602\):
\[ \text{pKa} \approx 6 - 0.602 = 5.398 \approx 5.4 \]

### Step 3: Use the Henderson-Hasselbalch equation to find the ratio.
We know the pH of the urine is 6.0. Plugging the values into the Henderson-Hasselbalch equation:
\[ 6.0 = 5.4 + \log\left(\frac{[\text{urate}]}{[\text{uric acid}]}\right) \]
Rearranging gives:
\[ \log\left(\frac{[\text{urate}]}{[\text{uric acid}]}\right) = 6.0 - 5.4 = 0.6 \]

### Step 4: Convert the logarithmic equation to exponential form.
To find the ratio, we convert from logarithmic form:
\[ \frac{[\text{urate}]}{[\text{uric acid}]} = 10^{0.6} \]
Calculating \(10^{0.6}\):
\[ 10^{0.6} \approx 3.98 \approx 4.0 \]

### Conclusion
The ratio of the concentration of urate ion to uric acid in the urine is approximately 4.0.

### Final Answer
(b) 4.0

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