Blood has pH of 7.40. Calculate the ratio of hydrogen carbonate ion `HCO_3^-` to `H_2CO_3` carbonic acid in blood to maintain its pH value. `pK_a(H_2CO_3)` = 6.37 .

Explanation

To calculate the ratio of hydrogen carbonate ion \((HCO_3^-)\) to carbonic acid \((H_2CO_3)\) in blood, we will use the Henderson-Hasselbalch equation, which is given by:

\[ \text{pH} = \text{pK}_a + \log\left(\frac{[HCO_3^-]}{[H_2CO_3]}\right) \]

### Step 1: Identify the known values
- pH of blood = 7.40
- \(pK_a\) of \(H_2CO_3\) = 6.37

### Step 2: Substitute the known values into the Henderson-Hasselbalch equation
\[ 7.40 = 6.37 + \log\left(\frac{[HCO_3^-]}{[H_2CO_3]}\right) \]

### Step 3: Rearrange the equation to isolate the logarithmic term
\[ \log\left(\frac{[HCO_3^-]}{[H_2CO_3]}\right) = 7.40 - 6.37 \]
\[ \log\left(\frac{[HCO_3^-]}{[H_2CO_3]}\right) = 1.03 \]

### Step 4: Convert the logarithmic expression to its exponential form
To eliminate the logarithm, we can rewrite the equation in exponential form:
\[ \frac{[HCO_3^-]}{[H_2CO_3]} = 10^{1.03} \]

### Step 5: Calculate \(10^{1.03}\)
Using a calculator, we find:
\[ 10^{1.03} \approx 10.671 \]

### Step 6: Write the final ratio
Thus, the ratio of hydrogen carbonate ion to carbonic acid in blood is approximately:
\[ \frac{[HCO_3^-]}{[H_2CO_3]} \approx 10.671 \]

### Conclusion
The ratio of hydrogen carbonate ion \((HCO_3^-)\) to carbonic acid \((H_2CO_3)\) in blood to maintain its pH value of 7.40 is approximately 10.671.

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