Problem 111 - Chemistry Matter and Change

Question

Apply Concepts Like all equilibrium constants, the value of \(K_{w}\) varies with temperature. \(K_{\mathrm{w}}\) equals \(2.92 \times 10^{-15}\) at \(10^{\circ} \mathrm{C}, 1.00 \times 10^{-14}\) at \(25^{\circ} \mathrm{C},\) and 2.92 \(\mathrm{x}\) \(10^{-14}\) at \(40^{\circ} \mathrm{C} .\) In light of this information, calculate and compare the pH values for pure water at these three temperatures. Based on your calculations, is it correct to say that the pH of pure water is always 7.0? Explain

Step-by-Step Solution

Answer

The pH of pure water is not always 7.0; it varies with temperature but remains neutral.

Step 1: Understanding Kw and pH

The ion product of water, \( K_w \), is the product of the concentrations of hydrogen ions \( [H^+] \) and hydroxide ions \( [OH^-] \) in pure water: \( K_w = [H^+][OH^-] \). Since water is neutral, \( [H^+] = [OH^-] \). Therefore, \( K_w = [H^+]^2 \). The pH is defined as \( -\log_{10}[H^+] \). So, to find the pH, we first need to find \( [H^+] \) by taking the square root of \( K_w \).

Step 2: Calculate pH at 10°C

At \( 10^{\circ}C \), \( K_w = 2.92 \times 10^{-15} \). Calculate \( [H^+] \) by \( [H^+] = \sqrt{K_w} = \sqrt{2.92 \times 10^{-15}} \). This gives \( [H^+] \approx 1.71 \times 10^{-7} \). The pH is then \( -\log_{10}(1.71 \times 10^{-7}) = 6.77 \).

Step 3: Calculate pH at 25°C

At \( 25^{\circ}C \), \( K_w = 1.00 \times 10^{-14} \). Calculate \( [H^+] = \sqrt{1.00 \times 10^{-14}} = 1.00 \times 10^{-7} \). The pH is \( -\log_{10}(1.00 \times 10^{-7}) = 7.0 \).

Step 4: Calculate pH at 40°C

At \( 40^{\circ}C \), \( K_w = 2.92 \times 10^{-14} \). Calculate \( [H^+] = \sqrt{2.92 \times 10^{-14}} = 1.71 \times 10^{-7} \). The pH is \( -\log_{10}(1.71 \times 10^{-7}) = 6.53 \).

Step 5: Analyze pH Values

The pH values calculated are 6.77 at \( 10^{\circ}C \), 7.0 at \( 25^{\circ}C \), and 6.53 at \( 40^{\circ}C \). These calculations show that the pH of pure water is not always 7.0; it varies with temperature. However, the water remains neutral as \( [H^+] = [OH^-] \) at all temperatures.

Key Concepts

pH Calculation

Calculating the pH is a fundamental aspect of understanding chemistry. The pH is a measure that tells us how acidic or basic a solution is on a scale from 0 to 14. Pure water, in its ideal state at room temperature, has a pH of 7 and is considered neutral.

When calculating the pH of a solution, we use the formula: \[\text{pH} = -\log_{10}[\text{H}^+]\]The \([\text{H}^+]\) symbol refers to the concentration of hydrogen ions in the solution. This might seem complicated at first, but the idea is to convert the concentration of hydrogen ions into an easy-to-understand number on a scale.

For example, if you calculate \([\text{H}^+]\) to be \(1 \times 10^{-7}\), the pH would be exactly 7, indicating neutrality.

Temperature Effect on pH

The temperature of a solution significantly impacts the pH level. As temperature changes, so does the autoionization of water, which is the process where water forms hydrogen and hydroxide ions. The equilibrium constant, \(K_w\), called the ion product of water, varies with temperature:

• At \(10^{\circ}C\), \(K_w = 2.92 \times 10^{-15}\)
• At \(25^{\circ}C\), \(K_w = 1.00 \times 10^{-14}\)
• At \(40^{\circ}C\), \(K_w = 2.92 \times 10^{-14}\)

It's important to note that as the temperature rises, \(K_w\) increases. This leads to a greater concentration of \([\text{H}^+]\) ions even though the solution may still be neutral because \([\text{H}^+] = [\text{OH}^-]\).

Higher temperatures reduce the pH of neutral water, even though fundamentally, it remains neutral.

Ion Product of Water

The ion product of water, or \(K_w\), is a central concept in understanding water chemistry. It represents the product of the concentrations of \([\text{H}^+]\) and \([\text{OH}^-]\) ions in pure water. For pure water, this relationship is expressed as:\[K_w = [\text{H}^+][\text{OH}^-]\]In neutral water, these concentrations are equal. This simplifies \(K_w\) into:\[K_w = [\text{H}^+]^2\]This equilibrium constant varies with temperature, and that is why the pH of water can change with temperature. However, the water remains neutral because the concentrations of \([\text{H}^+]\) and \([\text{OH}^-]\) adjust equally, maintaining the balance.

Understanding \(K_w\) is crucial in predicting how pH will shift due to temperature variations, demonstrating that while the numerical value of pH changes, the concept of neutrality is maintained.