Question

At the temperature of the human body, \(37^{\circ} \mathrm{C}\), the value of \(K_{\mathrm{w}}\) is \(2.5 \times 10^{-14} .\) Calculate \(\left[\mathrm{H}^{+}\right],\left[\mathrm{OH}^{-}\right], \mathrm{pH},\) and \(\mathrm{pOH}\) of pure water at this temperature. What is the relationship between \(\mathrm{pH}, \mathrm{pOH},\) and \(\mathrm{p} K_{\mathrm{w}}\) at this temperature? Is \(\mathrm{pH} 7.00\) water neutral at this temperature?

Short answer

At 37°C, \[\left[\mathrm{H}^{+}\right] = \left[\mathrm{OH}^{-}\right] = 5.0 \times 10^{-7}\ M\], \[\mathrm{pH} = \mathrm{pOH} = 6.30\], \[\mathrm{p}K_{\mathrm{w}} = 13.60\], so pH 7.00 is not neutral at this temperature; neutral pH is actually 6.80.

Step-by-step solution

Calculate Hydrogen and Hydroxide Ion Concentrations

For pure water at equilibrium, \[\mathrm{H}^{+}\]\ is equal to \[\mathrm{OH}^{-}\]\. Since \[K_{\mathrm{w}}\] is the ionic product of water, which is \[K_{\mathrm{w}} = [\mathrm{H}^{+}][\mathrm{OH}^{-}]\] at equilibrium, we can calculate the concentration of hydrogen and hydroxide ions by taking the square root of \[K_{\mathrm{w}}\]. \[\left[\mathrm{H}^{+}\right] = \left[\mathrm{OH}^{-}\right] = \sqrt{K_{\mathrm{w}}}\]

Calculate the Hydrogen Ion Concentration

Take the square root of \[K_{\mathrm{w}}\] to find the concentrations of \[\mathrm{H}^{+}\]\ and \[\mathrm{OH}^{-}\]\. \[\left[\mathrm{H}^{+}\right] = \left[\mathrm{OH}^{-}\right] = \sqrt{2.5 \times 10^{-14}}\]

Evaluate the Square Root

After calculating the square root, we get the concentrations: \[\left[\mathrm{H}^{+}\right] = \left[\mathrm{OH}^{-}\right] = 5.0 \times 10^{-7}\ M\]

Calculate the pH

To find the pH, use the formula \[\mathrm{pH} = -\log\left[\mathrm{H}^{+}\right]\]\ and plug in the concentration of hydrogen ions. \[\mathrm{pH} = -\log(5.0 \times 10^{-7})\]

Calculate the pOH

Similarly for pOH, use the formula \[\mathrm{pOH} = -\log\left[\mathrm{OH}^{-}\right]\]\ and use the concentration of hydroxide ions. \[\mathrm{pOH} = -\log(5.0 \times 10^{-7})\]

Determine the Relationship Between pH, pOH, and pKw

The relationship is defined by the equation \[\mathrm{pH} + \mathrm{pOH} = \mathrm{p}K_{\mathrm{w}}\], where \[\mathrm{p}K_{\mathrm{w}} = -\log\ K_{\mathrm{w}}\]. Calculate \[\mathrm{p}K_{\mathrm{w}}\] by applying the negative logarithm to the given \[K_{\mathrm{w}}\].

Is pH 7.00 Water Neutral at this Temperature?

At 37 degrees Celsius, if \[\mathrm{pH} + \mathrm{pOH}\] is equal to \[\mathrm{p}K_{\mathrm{w}}\]\ and the sum is not 14, then pH 7.00 is not neutral at this temperature. The neutral pH would be half of \[\mathrm{p}K_{\mathrm{w}}\] at this temperature.

Key concepts

Ionic Product of Water (Kw)

Understanding the ionic product of water, commonly represented as Kw, is fundamental in the study of aqueous solutions and their pH levels. It is defined by the equation
\[K_{w} = [H^{+}][OH^{-}]\]
The constant Kw represents the product of the molar concentrations of hydrogen ions (\( [H^{+}] \)) and hydroxide ions (\( [OH^{-}] \)) in water at equilibrium. At 25°C (room temperature), the value of Kw is typically \( 1 \times 10^{-14} \).
However, it's important to remember that Kw is temperature dependent; as the temperature increases, so does the value of Kw. This temperature dependence has crucial implications for pH calculations and understanding the acidity or basicity of a solution at different temperatures.

Hydrogen Ion Concentration

The hydrogen ion concentration in a solution, expressed as \( [H^{+}] \), directly affects the solution's pH value. pH is calculated using the negative base-10 logarithm of the hydrogen ion concentration:
\[ pH = -\text{log}([H^{+}]) \]
A higher concentration of hydrogen ions corresponds to a lower pH value, indicating a more acidic solution. Conversely, a lower concentration of hydrogen ions indicates a higher pH value, signifying a more basic or alkaline solution.
The concentration of hydrogen ions is determined by both the substances dissolved in the solution and the temperature, because of the temperature's effect on the dissociation of water molecules.

Hydroxide Ion Concentration

Similarly, the hydroxide ion concentration (\( [OH^{-}] \)) is the other key factor in determining the pH of a solution. It can be calculated using the relationship provided by the ionic product of water, Kw:
\[ [OH^{-}] = \frac{K_{w}}{[H^{+}]} \]
Therefore, knowing either the hydrogen or hydroxide ion concentration helps you calculate the other. In the case of pure water or a neutral solution, \( [H^{+}] \) and \( [OH^{-}] \) are equal. The pOH can also be found using the equation:
\[ pOH = -\text{log}([OH^{-}]) \]
In the provided exercise, equal concentrations of \( [H^{+}] \) and \( [OH^{-}] \) were determined, which is characteristic of pure water.

Temperature Dependence of pH

The temperature dependence of pH is a vital consideration when examining solutions outside standard laboratory conditions. As temperature increases, Kw also increases, typically resulting in a higher concentration of hydrogen ions. This can lead to a decrease in pH, implying that the solution becomes more acidic at elevated temperatures.
The provided exercise illustrates this concept with a pH calculation at human body temperature (37°C), where the neutral pH is no longer 7.00 as it would be at standard room temperature. In similar temperature-variant scenarios, the neutral pH would be redefined as the pH where the concentrations of \( [H^{+}] \) and \( [OH^{-}] \) remain equal, which would be expressed as half of \( pK_{w} \) at that temperature.
The relationship between pH and temperature is an excellent example of the necessity to consider environmental conditions when studying chemical equilibrium and acidity in biological, chemical, and environmental contexts.