Question

The average \(\mathrm{pH}\) of normal arterial blood is \(7.40\). At normal body temperature \(\left(37^{\circ} \mathrm{C}\right), K_{w}=2.4 \times 10^{-14} \cdot\) Calculate \(\left[\mathrm{H}^{+}\right],\left[\mathrm{OH}^{-}\right]\), and \(\mathrm{pOH}\) for blood at this temperature.

Short answer

The concentration of hydronium ions in normal arterial blood at 37°C is \(\mathrm{[H^+]} = 3.98 \times 10^{-8} \mathrm{M}\), the concentration of hydroxide ions is \(\mathrm{[OH^{-}]} = 6.03 \times 10^{-7} \mathrm{M}\), and the pOH is 6.22.

Step-by-step solution

Find the hydronium ion concentration (\(\mathrm{[H^+]}\))

We are given the pH of the blood, so we can use the definition of the pH to find the concentration of hydronium ions: \(\mathrm{pH} = -\log(\mathrm{[H^+]})\) Rearranging the equation, we get: \(\mathrm{[H^+]} = 10^{-\mathrm{pH}}\) Plugging the given pH value of 7.40, we calculate the hydronium ion concentration: \(\mathrm{[H^+]} = 10^{-7.40} = 3.98 \times 10^{-8} \mathrm{M}\)

Find the hydroxide ion concentration (\(\mathrm{[OH^{-}]}\))

To find the hydroxide ion concentration, we will use the ion product of water, \(K_w\), at 37°C: \(K_w = \mathrm{[H^+]} \cdot \mathrm{[OH^{-}]}\) We can rearrange the equation to find the hydroxide ion concentration: \(\mathrm{[OH^{-}]} = \frac{K_w}{\mathrm{[H^+]}}\) Plugging in the given value of \(K_w = 2.4 \times 10^{-14}\) and the calculated value of \(\mathrm{[H^+]} = 3.98 \times 10^{-8}\mathrm{M}\), we calculate the hydroxide ion concentration: \(\mathrm{[OH^{-}]} = \frac{2.4 \times 10^{-14}}{3.98 \times 10^{-8}} = 6.03 \times 10^{-7} \mathrm{M}\)

Calculate pOH for blood at this temperature

Now that we have calculated the hydroxide ion concentration, we can use the definition of pOH to find the pOH value: \(\mathrm{pOH} = -\log(\mathrm{[OH^{-}]})\) Plugging in the calculated value of \(\mathrm{[OH^{-}]} = 6.03 \times 10^{-7} \mathrm{M}\), we get: \(\mathrm{pOH} = -\log{(6.03 \times 10^{-7})} = 6.22\) Therefore, the concentration of hydronium ions in normal arterial blood at 37°C is \(\mathrm{[H^+]} = 3.98 \times 10^{-8} \mathrm{M}\), the concentration of hydroxide ions is \(\mathrm{[OH^{-}]} = 6.03 \times 10^{-7} \mathrm{M}\), and the pOH is 6.22.

Key concepts

Hydronium Ion Concentration

Understanding hydronium ion concentration is pivotal in the study of acid-base chemistry. Hydronium ions, denoted as \(\mathrm{[H^+]}\), are essentially hydrogen ions combined with water molecules, resulting in \(\mathrm{H_3O^+}\).

The pH scale measures the acidity or basicity of a solution and is directly related to the concentration of hydronium ions. The pH is calculated as the negative logarithm of the hydronium ion concentration: \(\mathrm{pH} = -\log(\mathrm{[H^+]})\).

To find \(\mathrm{[H^+]}\) from a given pH, we invert the process, resulting in \(\mathrm{[H^+]} = 10^{-\mathrm{pH}}\). For instance, with a pH of 7.40, the hydronium ion concentration in arterial blood becomes \(\mathrm{[H^+]} = 10^{-7.40} = 3.98 \times 10^{-8} \mathrm{M}\).

This concentration indicates the blood's ability to neutralize acids, which is essential in maintaining homeostasis. High \(\mathrm{[H^+]}\) levels can lead to acidosis, while low levels can lead to alkalosis.

Hydroxide Ion Concentration

The measurement of hydroxide ion concentration, shown as \(\mathrm{[OH^-]}\), is equally crucial in acid-base chemistry. These ions are the counterparts to hydronium ions in aqueous solutions.

The concentration of \(\mathrm{[OH^-]}\) also influences the pH and is related to the hydronium ion concentration via the ion product of water (\(K_w\)), which at a specific temperature, is a constant. The relationship is expressed as \(\mathrm{[H^+]} \cdot \mathrm{[OH^-]} = K_w\).

At a known \(K_w\) and given \(\mathrm{[H^+]}\), hydroxide ion concentration can be determined: \(\mathrm{[OH^-]} = \frac{K_w}{\mathrm{[H^+]}\). Using \(K_w = 2.4 \times 10^{-14}\) at 37°C and \(\mathrm{[H^+]} = 3.98 \times 10^{-8}\mathrm{M}\), the hydroxide ion concentration in blood is \(\mathrm{[OH^-]} = 6.03 \times 10^{-7} \mathrm{M}\).

Understanding \(\mathrm{[OH^-]}\) is key in environmental science, water treatment processes, and maintaining the delicate balance in biological systems.

Ion Product of Water

The ion product of water (\(K_w\)) is a fundamental constant in chemistry that relates to the inherent dissociation of water into hydronium and hydroxide ions. At 25°C, \(K_w\) is \(1.0 \times 10^{-14}\), but it varies with temperature.

At body temperature (37°C), \(K_w\) increases to \(2.4 \times 10^{-14}\), indicating more ionization compared to cooler temperatures. To find either hydronium or hydroxide ion concentration, we rearrange the expression: \(\mathrm{[OH^-]} = \frac{K_w}{\mathrm{[H^+]}\) or \(\mathrm{[H^+]} = \frac{K_w}{\mathrm{[OH^-]}\), given the other is known.

In practical terms, \(K_w\) is instrumental in applications ranging from aquatic life sustainability to the pharmaceutical function as it governs the relationship between \(\mathrm{[H^+]}\) and \(\mathrm{[OH^-]}\) in any given aqueous solution.

Blood pH Regulation

Blood pH regulation is critical for human health, with a narrow optimal range between 7.35 and 7.45. Slight deviations can have significant consequences on bodily functions.

Various mechanisms are in place to regulate blood pH, including buffer systems, respiration, and renal function. Buffers quickly neutralize excess acids or bases, while respiration adjusts the pH by altering carbon dioxide levels. Renal regulation involves the kidney's ability to excrete or retain hydrogen and bicarbonate ions.

Understanding blood pH regulation is fundamental in the medical field, particularly in the diagnosis and treatment of metabolic or respiratory acidosis and alkalosis. Maintaining the precise balance of hydronium and hydroxide ions in the blood is a testament to the body's complex homeostatic capabilities.