Question

(a) What is the ratio of \(\mathrm{HCO}_{3}^{-}\) to \(\mathrm{H}_{2} \mathrm{CO}_{3}\) in blood of pH \(7.4\) ? (b) What is the ratio of \(\mathrm{HCO}_{3}^{-}\) to \(\mathrm{H}_{2} \mathrm{CO}_{3}\) in an exhausted marathon runner whose blood pH is 7.1?

Short answer

(a) At pH 7.4 (normal blood pH), the ratio of \(\mathrm{HCO}_{3}^{-}\) to \(\mathrm{H}_{2} \mathrm{CO}_{3}\) is approximately 20. (b) At pH 7.1 (exhausted marathon runner's blood pH), the ratio of\(\mathrm{HCO}_{3}^{-}\) to \(\mathrm{H}_{2} \mathrm{CO}_{3}\) is 10.

Step-by-step solution

Use the Henderson-Hasselbalch equation for pH 7.4

For normal blood pH of 7.4, we have: $$7.4 = pK_a + \log\frac{[\mathrm{HCO}_{3}^{-}]}{[\mathrm{H}_{2} \mathrm{CO}_{3}]}$$ We need to solve for the ratio \( \frac{[\mathrm{HCO}_{3}^{-}]}{[\mathrm{H}_{2} \mathrm{CO}_{3}]}\).

Rearrange the equation for the desired ratio

First, subtract \(pK_a\) from both sides of the equation: $$7.4 - pK_a = \log\frac{[\mathrm{HCO}_{3}^{-}]}{[\mathrm{H}_{2} \mathrm{CO}_{3}]}$$ Now we will use the inverse log function (anti-log or 10^x) to isolate the ratio: $$\frac{[\mathrm{HCO}_{3}^{-}]}{[\mathrm{H}_{2} \mathrm{CO}_{3}]} = 10^{7.4 - pK_a}$$

Determine the ratio of \(\mathrm{HCO}_{3}^{-}\) to \(\mathrm{H}_{2} \mathrm{CO}_{3}\) for pH 7.4

In this step, we will use the value of \(pK_a\) for the bicarbonate ion, which is approximately 6.1. Plugging this value into the equation from step 2: $$\frac{[\mathrm{HCO}_{3}^{-}]}{[\mathrm{H}_{2} \mathrm{CO}_{3}]} = 10^{7.4 - 6.1} = 10^{1.3} \approx 20$$ So, at pH 7.4, the ratio of \(\mathrm{HCO}_{3}^{-}\) to \(\mathrm{H}_{2} \mathrm{CO}_{3}\) is about 20.

Use the Henderson-Hasselbalch equation for pH 7.1

For the marathon runner's blood pH of 7.1, we have: $$7.1 = pK_a + \log\frac{[\mathrm{HCO}_{3}^{-}]}{[\mathrm{H}_{2} \mathrm{CO}_{3}]}$$ We need to solve for the ratio \( \frac{[\mathrm{HCO}_{3}^{-}]}{[\mathrm{H}_{2} \mathrm{CO}_{3}]}\) in this case as well.

Determine the ratio of \(\mathrm{HCO}_{3}^{-}\) to \(\mathrm{H}_{2} \mathrm{CO}_{3}\) for pH 7.1

Using the same method as before, we'll subtract \(pK_a\) from both sides and use the inverse log function: $$\frac{[\mathrm{HCO}_{3}^{-}]}{[\mathrm{H}_{2} \mathrm{CO}_{3}]} = 10^{7.1 - 6.1} = 10^{1} = 10$$ So, at pH 7.1, the ratio of \(\mathrm{HCO}_{3}^{-}\) to \(\mathrm{H}_{2} \mathrm{CO}_{3}\) is 10.

Conclusion

To summarize: (a) At pH 7.4 (normal blood pH), the ratio of \(\mathrm{HCO}_{3}^{-}\) to \(\mathrm{H}_{2} \mathrm{CO}_{3}\) is approximately 20. (b) At pH 7.1 (exhausted marathon runner's blood pH), the ratio of\(\mathrm{HCO}_{3}^{-}\) to \(\mathrm{H}_{2} \mathrm{CO}_{3}\) is 10.

Key concepts

Blood pH Balance

The human body meticulously maintains the pH level of blood, keeping it slightly alkaline at around 7.4, a state crucial for optimal cellular function and overall health. The pH scale ranges from 0 to 14, with 7 being neutral. Any deviation from the normal pH range can have significant consequences, leading to a condition known as acidosis if blood becomes too acidic or alkalosis if it becomes too basic. To prevent such imbalances, the body employs buffer systems which are compounds that resist changes in pH by absorbing excess hydrogen ions (H⁺) or donating them when necessary.

One of the main buffering systems is the bicarbonate buffer system, which involves a delicate balance between bicarbonate ions (HCO₃^-) and carbonic acid (H₂CO₃). This system is regulated through respiratory and renal pathways to ensure that the blood pH remains within the narrow range that is vital for life.

Acid-Base Equilibrium

Acid-base equilibrium in the body is a complex process aiming for balance between the production and elimination of acids and bases. This equilibrium is necessary to maintain stable physiological functions and metabolic processes. The main elements of this equilibrium include the bicarbonate buffer system, the respiratory system, and the renal system.

The lungs control the amount of carbon dioxide (CO₂), which is part of the carbonic acid equation: CO₂ + H₂O ↔ H₂CO₃ ↔ HCO₃^- + H⁺. By altering breathing rate and depth, the lungs can increase or decrease the elimination of CO₂, shifting this equilibrium and changing the pH of blood. Similarly, the kidneys manage the excretion or retention of bicarbonate (HCO₃^-) and hydrogen ions to maintain the correct pH balance.

Bicarbonate Buffer System

The bicarbonate buffer system is an essential part of maintaining the body's pH balance. It is made up of bicarbonate ions (HCO₃^-) and carbonic acid (H₂CO₃), and it regulates blood pH by neutralizing excess acids and bases.

The Henderson-Hasselbalch equation, which relates the pH of a buffer solution to the concentration ratio of the buffer components, is fundamental in understanding how this system works. The equation is written as:
\[ pH = pK_a + \text{log} \left ( \frac{[HCO_{3}^{-}]}{[H_2CO_3]} \right ) \]
where pK_a is the acid dissociation constant of carbonic acid. This equation is particularly useful in clinical settings to assess the acid-base status of patients and can guide the diagnosis and treatment of metabolic and respiratory disorders. For example, an increased HCO₃^- to H₂CO₃ ratio indicates alkalosis, while a decreased ratio corresponds to acidosis. As demonstrated in the exercise solution, this equation makes it possible to calculate the changes in the bicarbonate buffer system under different conditions, such as the physiological response to marathon running.