Question

In humans, blood \(\mathrm{pH}\) is maintained within a narrow range: acidosis occurs if the blood \(\mathrm{pH}\) is below \(7.35,\) and alkalosis occurs if the \(\mathrm{pH}\) is above 7.45 . Given that the \(\mathrm{p} K_{\mathrm{w}}\) of blood is 13.63 at \(37^{\circ} \mathrm{C}\) (body temperature), what is the normal range of \(\left[\mathrm{H}_{3} \mathrm{O}^{+}\right]\) and of \(\left[\mathrm{OH}^{-}\right]\) in blood?

Short answer

Normal \( [\text{H}_3\text{O}^+] \) range: 3.55 to 4.47 \times 10^{-8} \text{M}, Normal \( [\text{OH}^-] \) range: 5.24 to 6.59 \times 10^{-7} \text{M}.

Step-by-step solution

- Identify the pH range for normal and abnormal conditions

Normal blood pH ranges from 7.35 to 7.45. Acidosis occurs when pH is below 7.35, and alkalosis occurs when pH is above 7.45.

- Define the relationship between pH and hydronium ion concentration \(\text{[}\text{H}_3\text{O}^+\text{]}\text{)}

The pH is defined by the equation \( \text{pH} = -\text{log}[\text{H}_3\text{O}^+] \). To find the concentration of \( [\text{H}_3\text{O}^+] \), use the formula \( [\text{H}_3\text{O}^+] = 10^{-\text{pH}} \).

- Calculate the concentration of \( [\text{H}_3\text{O}^+] \) for lower and upper pH limits

For pH = 7.35: \[ [\text{H}_3\text{O}^+] = 10^{-7.35} \approx 4.47 \times 10^{-8} \text{M} \] For pH = 7.45: \[ [\text{H}_3\text{O}^+] = 10^{-7.45} \approx 3.55 \times 10^{-8} \text{M} \]

- Determine the relationship between \( [\text{H}_3\text{O}^+] \) and \( [\text{OH}^-] \)

Use the ion product constant \( \text{K}_\text{w} \), defined by the equation \[ [\text{H}_3\text{O}^+][\text{OH}^-] = 10^{-\text{p}\text{K}_\text{w}} = 10^{-13.63} \] This simplifies to \text{K}_\text{w} = 2.34 \times 10^{-14} \text{M}^2 \.

- Calculate \( [\text{OH}^-] \) using the pH limits

For pH = 7.35: \ [\text{H}_3\text{O}^+] \times [\text{OH}^-] = \text{K}_\text{w} \ \( 4.47 \times 10^{-8} \text{M} \times [\text{OH}^-] = 2.34 \times 10^{-14} \text{M}^2 \) \[ [\text{OH}^-] \approx 5.24 \times 10^{-7} \text{M} \] For pH = 7.45: \ [\text{H}_3\text{O}^+] \times [\text{OH}^-] = \text{K}_\text{w} \ \( 3.55 \times 10^{-8} \text{M} \times [\text{OH}^-] = 2.34 \times 10^{-14} \text{M}^2 \) \[ [\text{OH}^-] \approx 6.59 \times 10^{-7} \text{M} \]

Key concepts

Acidosis

Acidosis is a condition where the blood pH drops below the normal range.
Normal blood pH ranges from 7.35 to 7.45.
When blood becomes too acidic, dropping below 7.35, this condition is termed as acidosis.
Acidosis can occur due to various factors, including kidney failure, respiratory conditions, or metabolic issues.
It is important to monitor and manage this condition as it can lead to severe health issues if left unchecked.

Alkalosis

Alkalosis is the opposite of acidosis; it occurs when blood pH rises above 7.45.
If the blood becomes too alkaline, it is termed as alkalosis.
Possible causes include hyperventilation, certain medications, or metabolic conditions.
Like acidosis, alkalosis requires proper attention to prevent complications.

Hydronium Ion Concentration

Hydronium ion concentration \(\left[\text{H}_3\text{O}^+\right]\) is directly related to pH.
pH is defined by the formula \(\text{pH} = -\log{\left[\text{H}_3\text{O}^+\right]}\).
To find the hydronium ion concentration from pH, use: \(\left[\text{H}_3\text{O}^+\right] = 10^{-\text{pH}}\).
Normal blood pH results in \(\left[\text{H}_3\text{O}^+\right]\) between approximately \(4.47 \times 10^{-8}\) M (for pH 7.35) and \(3.55 \times 10^{-8}\) M (for pH 7.45).

Ion Product Constant

The ion product constant, denoted as \(K_w\), relates the concentrations of hydronium and hydroxide ions in water.
At 37°C, \(K_w\) for blood is given as \(10^{-13.63}\), or \(2.34 \times 10^{-14} \text{M}^2\).
This equation helps in determining the relationship between \(\left[\text{H}_3\text{O}^+\right]\) and \(\left[\text{OH}^-\right]\): \[\left[\text{H}_3\text{O}^+\right] \left[\text{OH}^-\right] = K_w\].
By knowing \(\left[\text{H}_3\text{O}^+\right]\), you can calculate \(\left[\text{OH}^-\right]\) and vice versa.

pH Calculation

To calculate pH, use the hydronium ion concentration formula.
The formula is \(\text{pH} = -\log{\left[\text{H}_3\text{O}^+\right]}\).
For normal blood, pH values range from 7.35 to 7.45.
You can convert these pH values to hydronium ion concentrations, \(4.47 \times 10^{-8} \text{M}\) for pH 7.35 and \(3.55 \times 10^{-8} \text{M}\) for pH 7.45.
These values are crucial for maintaining healthy bodily functions and ensuring optimal metabolic and physiological processes.