Problem 81 - Thomas Calculus

Question

In any solution, the product of the hydronium ion concentration \(\left[\mathrm{H}_{3} \mathrm{O}^{+}\right]\) (moles \(/ \mathrm{L} )\) and the hydroxyl ion concentration \(\left[\mathrm{OH}^{-}\right]\) (moles \(/ \mathrm{L} )\) is about \(10^{-14} .\) a. What value of \(\left[\mathrm{H}_{3} \mathrm{O}^{+}\right]\) minimizes the sum of the concentrations, \(S=\left[\mathrm{H}_{3} \mathrm{O}^{+}\right]+\left[\mathrm{OH}^{-}\right] ?\) (Hint: Change notation. Let \(x=\left[\mathrm{H}_{3} \mathrm{O}^{+}\right] . )\) b. What is the pH of a solution in which \(S\) has this minimum value? c. What ratio of \(\left[\mathrm{H}_{3} \mathrm{O}^{+}\right]\) to \(\mathrm{OH}^{-} ]\) minimizes \(S ?\)

Step-by-Step Solution

Answer

a. \([\mathrm{H}_3\mathrm{O}^+] = 10^{-7}\) M b. pH = 7 c. The ratio is 1:1.

Step 1: Understand the relationship between hydronium and hydroxide ions

According to the problem, the product of the hydronium ion concentration \( [\mathrm{H}_3\mathrm{O}^+] \) and the hydroxide ion concentration \( [\mathrm{OH}^-] \) is always approximately \( 10^{-14} \) at 25°C. This relationship is known as the ion-product constant of water, expressed as:\[[\mathrm{H}_3\mathrm{O}^+][\mathrm{OH}^-] = 10^{-14}\]

Step 2: Express the sum of concentrations in terms of one variable

We need to find \([\mathrm{H}_3\mathrm{O}^+]\) that minimizes the sum \( S = [\mathrm{H}_3\mathrm{O}^+] + [\mathrm{OH}^-] \). First, express \([\mathrm{OH}^-]\) using the product relationship:\[[\mathrm{OH}^-] = \frac{10^{-14}}{[\mathrm{H}_3\mathrm{O}^+]}\]Now, rewrite the sum S using \( x = [\mathrm{H}_3\mathrm{O}^+] \):\[S = x + \frac{10^{-14}}{x}\]

Step 3: Find the derivative to minimize the sum

Minimize the function \( S(x) = x + \frac{10^{-14}}{x} \). Calculate the derivative \( S'(x) \):\[S'(x) = 1 - \frac{10^{-14}}{x^2}\]Set the derivative equal to zero to find the critical points:\[1 - \frac{10^{-14}}{x^2} = 0\]

Step 4: Solve for critical points

Solving \(1 - \frac{10^{-14}}{x^2} = 0\) gives:\[\frac{10^{-14}}{x^2} = 1 \x^2 = 10^{-14} \x = 10^{-7}\]Thus, \([\mathrm{H}_3\mathrm{O}^+] = 10^{-7}\) M is the value that minimizes S.

Step 5: Calculate the pH value

The pH is calculated using the hydronium ion concentration:\[pH = -\log_{10}[\mathrm{H}_3\mathrm{O}^+]\]Substitute \([\mathrm{H}_3\mathrm{O}^+] = 10^{-7}\):\[pH = -\log_{10}(10^{-7}) = 7\]Thus, the pH of the solution is 7.

Step 6: Determine the ratio of hydronium to hydroxide ions

From Step 4, we have \([\mathrm{H}_3\mathrm{O}^+] = 10^{-7}\) and using the product constant,\[[\mathrm{OH}^-] = \frac{10^{-14}}{10^{-7}} = 10^{-7}\]The ratio \([\mathrm{H}_3\mathrm{O}^+]:[\mathrm{OH}^-]\) is:\[1:1\]

Key Concepts

Ion Product of Water

The ion product of water is a fundamental concept in chemistry that helps us understand how water ionizes into hydronium and hydroxide ions. This ionization is a weak process, meaning that spontaneous water ionization happens only to a tiny extent. However, at 25°C, the product of the concentrations of these ions remains constant: \[ [H_3O^+][OH^-] = 10^{-14} \]This expression is known as the ion-product constant of water, often referred to as \( K_w \). It is crucial in calculations involving pH and pOH, which describe the acidity or basicity of a solution. This constant value means that if the concentration of hydronium ions increases, the concentration of hydroxide ions must decrease proportionally, and vice versa, to keep the product at \( 10^{-14} \).
Understanding this balance is key when studying acid-base equilibrium, as it sets the foundation for calculating pH and analyzing equilibrium in aqueous solutions.

pH Calculation

The concept of pH is integral in chemistry when discussing the acidity or basicity of a solution. The pH scale measures the concentration of hydronium ions \([H_3O^+]\) in a solution, and is calculated using the formula:\[ pH = -\log_{10}[H_3O^+] \]A neutral solution, like pure water, has a pH of 7, meaning the concentration of \([H_3O^+]\) is \(10^{-7}\) mol/L. The pH scale typically ranges from 0 to 14, where a pH less than 7 indicates an acidic solution and greater than 7 indicates a basic solution.
To find the pH of a solution where you already know the concentration of \([H_3O^+]\), simply apply this formula. Conversely, if you know the pH, you can determine \([H_3O^+]\) using the inverse operation. This relationship helps chemists understand the conditions of a solution, particularly when determining reaction rates or optimizing reaction conditions, as slight changes in pH can significantly affect chemical processes.

Optimization in Chemistry

Optimization in chemistry often involves finding conditions that maximize or minimize something, like concentration or reaction rate. In this exercise, the aim was to minimize the sum of hydronium and hydroxide ion concentrations to achieve an optimal balance. To do this, we use calculus, specifically by finding the derivative of the function describing the sum of concentrations:\[ S(x) = x + \frac{10^{-14}}{x} \]Finding the derivative \( S'(x) \) helps identify critical points where the function's value could be minimized or maximized. Here, by setting \( S'(x) = 0 \), we determine at what point the sum \( S \) is minimized. Calculations show the minimal value is achieved when \([H_3O^+] = 10^{-7}\) mol/L, which mirrors the neutral pH condition in water.
Such optimization techniques are not limited to this context but are widely used across different chemistry fields, including synthetic chemistry, where reaction optimization can lead to better yields or more efficient processes.