The dissociation of water into H 3 O + and OH – ions depends on temperature. At 0 °C the [H 3 O + ] = 3.38 x 10 –8 M , at 25 °C the [H 3 O + ] = 1.00 x 10 –7 M , and at 50 °C the [H 3 O + ] = 2.34 x 10 –7 M . b. What is the value of K w at 0 °C and 50 °C?

Hey, everyone, we're told that temperature affects the dissociation of water into hydro ni um and hydroxide ions given the moralities of hydro knee um ions at five degrees Celsius, 25 degrees Celsius and 45 degrees Celsius. Calculate the values of K W in three significant figures at five degrees Celsius and at 45 degrees Celsius. Now, at five degrees Celsius, the concentration of our hydro knee um ion is equal to 4.32 times 10 to the negative eight mol per kilogram of water. And at 25 degrees Celsius, the concentration of hydro ni um ion is equal to one point oh times 10 to the negative seven mol per kilogram of water. And at 45 degrees Celsius, the concentration of our hydro ni um ion is equal to 1.97 times 10 to the negative seven mol per kilogram of water. First, let's go ahead and write the dissociation of water, which is going to be two of our water which dissociates into our hydro ni um ion plus our hydroxide ion. Now, we know that our K W is equal to the concentration of our hydro knee. Um ion times the concentration of our hydroxide ion. Now, since the concentration of our hydro ni um ion is equal to the concentration of our hydroxide ion. We can then say that our K W is equal to the concentration of our hydro knee um ion squared. Now that we've written this out, let's go ahead and plug in our values. So at five degrees Celsius, we can go ahead and solve for our K W which is the concentration of our hydro ni um ion squared. So at five degrees Celsius, we were told that this is 4.32 times 10 to the negative eight mole per kilogram of water. So we're going to square this and this will get us to a value of 1.87 times 10 to the negative 15 for our K W And this is in three significant figures. Now, let's go ahead and solve this at 45°C. Again, our K W is going to be the concentration of our hydro ni um ion squared. And we were told that this is 1.97 times 10 to the negative seven mol per kilogram of water at 45 degrees Celsius. And we're going to square this. So when we calculate this out, we end up with a KW of 3.88 times 10 to the negative 14, which is going to be our final answer. Now, I hope this made sense and let us know if you have any questions.

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10. Acids and Bases

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10. Acids and Bases

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Problem 99b

Textbook Question

The dissociation of water into H₃O⁺ and OH^– ions depends on temperature. At 0 °C the [H₃O⁺] = 3.38 x 10^–8 M, at 25 °C the [H₃O⁺] = 1.00 x 10^–7 M, and at 50 °C the [H₃O⁺] = 2.34 x 10^–7 M.
b. What is the value of K_w at 0 °C and 50 °C?

Verified step by step guidance

Step 1: Recall the expression for the ionization constant of water, K_w, which is defined as the product of the concentrations of hydronium ions ([H₃O⁺]) and hydroxide ions ([OH⁻]). Mathematically, K_w = [H₃O⁺] × [OH⁻].

Step 2: Understand that water is neutral, meaning the concentrations of [H₃O⁺] and [OH⁻] are equal at any given temperature. Therefore, [OH⁻] = [H₃O⁺].

Step 3: Substitute the given values of [H₃O⁺] at 0 °C and 50 °C into the formula for K_w. For 0 °C, [H₃O⁺] = 3.38 × 10⁻⁸ M, so K_w = (3.38 × 10⁻⁸ M) × (3.38 × 10⁻⁸ M). For 50 °C, [H₃O⁺] = 2.34 × 10⁻⁷ M, so K_w = (2.34 × 10⁻⁷ M) × (2.34 × 10⁻⁷ M).

Step 4: Perform the multiplication for each temperature to find the value of K_w. For 0 °C, multiply (3.38 × 10⁻⁸) by itself. For 50 °C, multiply (2.34 × 10⁻⁷) by itself.

Step 5: Express the final values of K_w for both temperatures in scientific notation, ensuring the units are in M² (molar squared) since K_w is a product of two molar concentrations.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Dissociation of Water

The dissociation of water refers to the process where water (H₂O) separates into hydronium ions (H₃O⁺) and hydroxide ions (OH⁻). This equilibrium is crucial for understanding acid-base chemistry and is influenced by temperature, as the concentrations of these ions change with thermal conditions.

Ion Product of Water (K_w)

The ion product of water, denoted as K_w, is the equilibrium constant for the dissociation of water. It is defined as the product of the concentrations of H₃O⁺ and OH⁻ ions in pure water. K_w varies with temperature, and its value at 25 °C is 1.0 x 10⁻¹⁴, serving as a reference point for calculating ion concentrations at different temperatures.

Temperature Dependence of K_w

K_w is temperature-dependent, meaning its value changes with temperature variations. As temperature increases, the dissociation of water becomes more favorable, leading to higher concentrations of H₃O⁺ and OH⁻ ions, thus increasing K_w. Understanding this relationship is essential for calculating K_w at different temperatures, such as 0 °C and 50 °C.

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