Problem 29 - Chemistry An Atoms First Approach

Question

Cream of tartar, a common ingredient in cooking, is the common name for potassium bitartrate (abbreviated KBT, molar mass \(=188.2 \mathrm{g} / \mathrm{mol}\) ). Historically, KBT was a crystalline solid that formed on the casks of wine barrels during the fermentation process. Calculate the maximum mass of KBT that can dissolve in \(250.0 \mathrm{mL}\) of solution to make a saturated solution. The \(K_{\mathrm{sp}}\) value for \(\mathrm{KBT}\) is \(3.8 \times 10^{-4}\).

Step-by-Step Solution

Answer

The maximum mass of potassium bitartrate (KBT) that can dissolve in 250.0 mL of solution to make a saturated solution is approximately \(0.919 \mathrm{g}\).

Step 1: 1. Write the balanced chemical equation for KBT dissolving in water.

The dissolution of potassium bitartrate (KBT) in water can be represented by the following balanced chemical equation: \[ \mathrm{KBT} \rightleftharpoons \mathrm{K^+} + \mathrm{Tartrate^{-}}\]

Step 2: 2. Determine the relationship between the concentration of the ions and Ksp.

According to the balanced chemical equation, 1 mole of KBT dissociates into 1 mole of K+ ions and 1 mole of Tartrate ions. Thus, if x moles of KBT dissolve in water, the concentration of K+ ions will be x, and the concentration of Tartrate ions will also be x.

Step 3: 3. Calculate the concentration of K+ and Tartrate ion in terms of x.

The concentration of the ions in the solution can be expressed in terms of x: \[ [\mathrm{K^+}] = x \] \[ [\mathrm{Tartrate^{-}}] = x \]

Step 4: 4. Write the expression for Ksp.

The expression for the solubility product constant, Ksp, can be written as the product of the concentrations (in moles) of K+ and Tartrate ions raised to their stoichiometric coefficients. \[ K_{\mathrm{sp}} = [\mathrm{K^+}] \cdot [\mathrm{Tartrate^{-}}] \]

Step 5: 5. Solve for x using the Ksp value given.

Now, substitute the known values into the Ksp expression and solve for x: \[3.8 \times 10^{-4} = x \cdot x\] \[ x^2 = 3.8 \times 10^{-4}\] \[ x = \sqrt{3.8 \times 10^{-4}} = 1.95 \times 10^{-2} M \]

Step 6: 6. Calculate the maximum mass of KBT that can dissolve in 250.0 mL of the solution using the molar mass of KBT.

First, calculate the moles of KBT in 250.0 mL (0.250 L) of the solution: \[ \mathrm{moles\,of\,KBT} = x \times \mathrm{volume\,of\,solution\,in\,L} = (1.95 \times 10^{-2} M) \cdot (0.250 L) = 4.88 \times 10^{-3} \mathrm{mol} \] Then, use the molar mass of KBT (188.2 g/mol) to calculate the maximum mass of KBT that can dissolve in the solution: \[ \mathrm{mass\,of\,KBT} = \mathrm{moles\,of\,KBT} \times \mathrm{molar\,mass\,of\,KBT} = (4.88 \times 10^{-3} \mathrm{mol}) \times (188.2 \mathrm{g/mol}) = 0.919 \mathrm{g} \] So, the maximum mass of KBT that can dissolve in 250.0 mL of the solution is approximately 0.919 g.

Key Concepts

Potassium Bitartrate

Potassium bitartrate, often known as cream of tartar, is a white, crystalline powder that is frequently used in baking to stabilize egg whites and cream, and to create a creamy texture in desserts. Chemically, it is represented as \( \text{KHT} \) or \( \text{KC}_4\text{H}_5\text{O}_6 \). This compound finds its origins from the winemaking process as a natural by-product formed inside the barrels.
When dissolved in water, potassium bitartrate reaches equilibrium to produce potassium ions (\( \text{K}^+ \)) and tartrate ions (\( \text{Tartrate}^- \)). It is this ability to dissociate that makes it useful in various culinary and chemical applications. Understanding how much of it can dissolve in water at a certain temperature is crucial for accurately using it in recipes or scientific experiments.
Overall, potassium bitartrate's role is significant both in kitchens and chemical equations where solubility plays a key factor.

Ksp (Solubility Product Constant)

The Solubility Product Constant, denoted as \( K_{sp} \), is a key concept in understanding the solubility of sparingly soluble salts like potassium bitartrate. It expresses the extent to which a compound can dissolve to form a saturated solution.
For potassium bitartrate dissolving in water:

• The balanced equation is \( \text{KBT} \rightleftharpoons \text{K}^+ + \text{Tartrate}^- \).
• The \( K_{sp} \) expression becomes: \( K_{sp} = [\text{K}^+] \cdot [\text{Tartrate}^-] \).

The numerical value of \( K_{sp} \) provides insight into how much of the compound can dissolve before reaching saturation. For example, a \( K_{sp} \) value of \( 3.8 \times 10^{-4} \) indicates a rather limited solubility which is typical for ionic compounds of this nature. Calculating \( K_{sp} \) involves determining the concentration of the ions when no more of the solid dissolves in the solution.

Chemical Equilibrium

Chemical equilibrium occurs when the rate of the forward reaction equals the rate of the reverse reaction. For potassium bitartrate dissolving in water, an equilibrium is established between the undissolved solid and the ions in solution.
In this scenario:

• The forward reaction involves the dissolution of \( \text{KBT} \) into \( \text{K}^+ \) and \( \text{Tartrate}^- \) ions.
• The reverse reaction involves the recombination of these ions back into solid potassium bitartrate.

At equilibrium, the concentrations of \( \text{K}^+ \) and \( \text{Tartrate}^- \) ions remain constant, and are defined by the \( K_{sp} \). This means that any change in concentration, pressure, or temperature can shift the equilibrium according to Le Chatelier's principle, which may lead to more dissolution or precipitation of the solid.

Molar Mass Calculation

Molar mass is vital in translating moles into grams, allowing for practical measurement of a substance. The molar mass of potassium bitartrate is 188.2 g/mol. This value represents the mass of one mole of the compound and is a sum of the atomic masses of the elements present in the formula \( \text{KC}_4\text{H}_5\text{O}_6 \).
Calculating the mass of a dissolved compound starts by finding the moles dissolved using concentration and volume:

• Given volume = 0.250 L, concentration \( = 1.95 \times 10^{-2} M \),
• Moles of \( \text{KBT} = \text{Concentration} \times \text{Volume} = 4.88 \times 10^{-3} \text{ mol} \).

Finally, use the molar mass to find the mass:

• Mass of \( \text{KBT} = \text{Moles} \times \text{Molar Mass} = 0.919 \text{ g} \).

This step is crucial for determining how much of a compound will dissolve in a given solution, ensuring precise execution in both scientific and culinary practices.