Problem 29 - Chemistry The Central Science

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\(?\)

Step-by-Step Solution

Answer

(a) In blood with a pH of 7.4, the ratio of \(\mathrm{HCO}_{3}^{-}\) to \(\mathrm{H}_{2} \mathrm{CO}_{3}\) is approximately 20:1. (b) In the blood of an exhausted marathon runner with a pH of 7.1, the ratio of \(\mathrm{HCO}_{3}^{-}\) to \(\mathrm{H}_{2} \mathrm{CO}_{3}\) is 10:1.

Step 1: Understand the Henderson-Hasselbalch equation

The Henderson-Hasselbalch equation is a useful tool in understanding the relationship between pH, pOH, and the concentrations of acids and their conjugate bases: \[ pH = pK_{a} + log_{10}(\frac{[\mathrm{Base}]}{[\mathrm{Acid}]}) \] where \(pH\) is the pH of the buffer solution, \(pK_{a}\) is the negative logarithm of the acid dissociation constant (\(K_{a}\)), and \([\mathrm{Base}]/[\mathrm{Acid}]\) is the ratio of the conjugate base (in our case, \(\mathrm{HCO}_{3}^{-}\)) to the acid (in our case, \(\mathrm{H}_{2} \mathrm{CO}_{3}\)).

Step 2: Determine the pKa value

For the carbonic acid/bicarbonate buffer system, the pKa value is approximately 6.1. We will use this value in our calculations.

Step 3: Calculate the ratio of HCO3- to H2CO3 in blood of pH 7.4 (Part a)

Given the pH of 7.4, we can substitute the known values into the Henderson-Hasselbalch equation and solve for the ratio: \[ 7.4 = 6.1 + log_{10}(\frac{[\mathrm{HCO}_{3}^{-}]}{[\mathrm{H}_{2}\mathrm{CO}_{3}]}) \] First, isolate the logarithm term by subtracting the \(pK_{a}\) value from the pH value: \[ 1.3 = log_{10}(\frac{[\mathrm{HCO}_{3}^{-}]}{[\mathrm{H}_{2}\mathrm{CO}_{3}]}) \] Next, remove the logarithm by taking the antilog (base 10) of both sides: \[ 10^{1.3} = \frac{[\mathrm{HCO}_{3}^{-}]}{[\mathrm{H}_{2}\mathrm{CO}_{3}]} \]

Step 4: Calculate the ratio of HCO3- to H2CO3 in blood of pH 7.1 (Part b)

Repeat the process for the given pH of 7.1: \[ 7.1 = 6.1 + log_{10}(\frac{[\mathrm{HCO}_{3}^{-}]}{[\mathrm{H}_{2}\mathrm{CO}_{3}]}) \] Subtract the \(pK_{a}\) value from the pH value: \[ 1.0 = log_{10}(\frac{[\mathrm{HCO}_{3}^{-}]}{[\mathrm{H}_{2}\mathrm{CO}_{3}]}) \] Remove the logarithm by taking the antilog (base 10) of both sides: \[ 10^{1.0} = \frac{[\mathrm{HCO}_{3}^{-}]}{[\mathrm{H}_{2}\mathrm{CO}_{3}]} \]

Step 5: Present the final ratios

(a) In blood with a pH of 7.4, the ratio of \(\mathrm{HCO}_{3}^{-}\) to \(\mathrm{H}_{2} \mathrm{CO}_{3}\) is approximately \(10^{1.3} \approx 19.95\), or approximately 20:1. (b) In the blood of an exhausted marathon runner with a pH of 7.1, the ratio of \(\mathrm{HCO}_{3}^{-}\) to \(\mathrm{H}_{2} \mathrm{CO}_{3}\) is \(10^{1.0} = 10\), or 10:1.

Key Concepts

pH calculation

Calculating pH is a fundamental concept in chemistry that helps us understand the acidity or basicity of a solution. The pH scale, which ranges from 0 to 14, measures how acidic or basic a substance is. values below 7 indicate acidity, and values above 7 indicate basicity.
The pH value is calculated using the concentration of hydrogen ions \([\mathrm{H}^+]\) in a solution:

• pH = -log_{10}([\mathrm{H}^+])

It is important to note that each unit change in pH represents a tenfold change in hydrogen ion concentration. Hence, a solution with a pH of 5 is 10 times more acidic than one with a pH of 6.
In buffer systems like the blood bicarbonate system, pH calculation becomes slightly more complex, as it involves the relative concentrations of the acid and its conjugate base. This is where the Henderson-Hasselbalch equation plays a crucial role, enabling us to find the ratio of the base to the acid and consequently gauge the system's pH.

buffer systems

Buffer systems are essential in maintaining the pH of a solution within a narrow range, even when acids or bases are added. They are crucial in many biological processes, keeping our bodies functioning correctly.
A buffer system consists of a weak acid and its conjugate base in equilibrium, or a weak base and its conjugate acid. This pair works by neutralizing excess hydrogen ions \([\mathrm{H}^+]\) or hydroxide ions \([\mathrm{OH}^-]\).
The bicarbonate buffer system in blood is a primary example, where carbonic acid \([\mathrm{H}_2\mathrm{CO}_3]\) acts as the weak acid and bicarbonate ion \([\mathrm{HCO}_3^-]\) as its conjugate base.

• When acids are added to the blood, the bicarbonate ion \([\mathrm{HCO}_3^-]\) reacts with excess \([\mathrm{H}^+]\) to form carbonic acid \([\mathrm{H}_2\mathrm{CO}_3]\), thus minimizing a drop in pH.
• Conversely, if the pH rises, carbonic acid dissociates to release hydrogen ions, helping to lower the pH.

Using the Henderson-Hasselbalch equation, we can determine the buffer capacity by calculating the ratio of the weak acid to its conjugate base, helping us predict the pH stability of the buffer system.

acid-base equilibrium

Acid-base equilibrium is the state of balance between the concentrations of acids and bases in a solution. These reactions are reversible, meaning they can proceed in both forward and backward directions.
Understanding this equilibrium helps predict how changes in concentration will affect the pH and the overall reaction direction.
In the case of the bicarbonate buffer system:

• The forward reaction \( \text{\[ \mathrm{H}_2\mathrm{CO}_3 \rightleftharpoons \mathrm{HCO}_3^- + \mathrm{H}^+\]} \) occurs as carbonic acid dissociates into bicarbonate and hydrogen ions.
• The reverse reaction takes place as bicarbonate ions react with hydrogen ions to form carbonic acid.

The balance of these reactions is sensitive to changes in pH, temperature, and concentration, making acid-base equilibrium complex yet fascinating.
Factors like buffer strength, the concentration of reactants, and environmental conditions can shift the equilibrium, resulting in compensated adjustments in a buffer system’s pH.
The Henderson-Hasselbalch equation is instrumental in these situations, allowing us to calculate the pH and examine how changes can disturb or restore equilibrium.