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The solubility of silver bromate, \(\mathrm{AgBrO}_{3}\), in water is \(0.0072 \mathrm{~g} / \mathrm{L}\). Calculate \(K_{s p .}\)

Short Answer

Expert verified
The solubility product constant, \( K_{sp} \), is \( 9.30 \times 10^{-10} \).

Step by step solution

01

Determine Molar Mass of Silver Bromate

To convert the solubility from grams per liter to molarity, first find the molar mass of silver bromate, AgBrO鈧. The molar mass is calculated as the sum of the atomic masses: Ag = 107.87 g/mol, Br = 79.904 g/mol, O = 16.00 g/mol (3 oxygens). So, the molar mass of AgBrO鈧 is 107.87 + 79.904 + (3 脳 16.00) = 235.764 g/mol.
02

Convert Solubility to Molarity

The solubility of AgBrO鈧 is given as 0.0072 g/L. To convert this to molarity (moles per liter), divide the solubility by the molar mass of AgBrO鈧. Molarity = (0.0072 g/L) / (235.764 g/mol) = 3.05 脳 10鈦烩伒 mol/L.
03

Write the Dissociation Equation

When AgBrO鈧 dissolves in water, it dissociates as follows: AgBrO鈧(s) 鈬 Ag鈦(aq) + BrO鈧冣伝(aq). This dissociation indicates that 1 mole of AgBrO鈧 produces 1 mole of Ag鈦 and 1 mole of BrO鈧冣伝.
04

Express the Concentrations

From the dissociation equation, the concentration of Ag鈦 ions and BrO鈧冣伝 ions in the solution will both equal the solubility of AgBrO鈧, which is 3.05 脳 10鈦烩伒 mol/L.
05

Calculate the Solubility Product Constant, K_{sp}

The solubility product constant, K_{sp}, is calculated using the concentrations of the ions:\[ K_{sp} = [Ag^+][BrO_3^-] = (3.05 脳 10^{-5})^2 = 9.30 脳 10^{-10} \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Molar Mass Calculation
The molar mass calculation is often the first step in solving problems related to solubility and reactions. For silver bromate ( AgBrO鈧), determining the molar mass requires summing the atomic masses of each element in the compound.
  • Silver ( Ag) has an atomic mass of 107.87 g/mol.
  • Bromine ( Br) is 79.904 g/mol.
  • Oxygen (O) has a mass of 16.00 g/mol, and there are three oxygens in the formula.
So, by adding these together: 107.87 + 79.904 + 3 脳 16.00, you get a molar mass of 235.764 g/mol for silver bromate.

This molar mass allows you to convert between grams and moles, a crucial factor in calculating molarity.
Ionic Dissociation
Understanding ionic dissociation is key to calculating concentrations in solutions. When silver bromate ( AgBrO鈧) dissolves in water, it breaks apart into ions:
  • AgBrO鈧(s) 鈬 Ag鈦(aq) + BrO鈧冣伝(aq)
This equation shows that each formula unit of silver bromate will dissociate into one silver ion ( Ag鈦) and one bromate ion ( BrO鈧冣伝).

This 1:1 stoichiometric ratio informs us that the concentrations of these ions in the solution will be the same as the solubility of AgBrO鈧. Such a balanced equation is vital for finding various chemical properties, like the solubility product constant (K_{sp}).
Molarity Conversion
Molarity ( M) is a way of expressing concentration, representing moles of solute per liter of solution. To convert from a given solubility in grams per liter to molarity, you'd need the compound's molar mass. For AgBrO鈧, already calculated as 235.764 g/mol, this conversion follows as:
  • Molarity = (0.0072 g/L) / (235.764 g/mol)
  • = 3.05 脳 10鈦烩伒 mol/L
Knowing how to convert solubility into molarity lets you calculate the concentration of ions. This information can be further applied to reaction stoichiometry and calculating equilibrium constants.
Solubility Equilibrium
Solubility equilibrium is the state reached when the dissolution and precipitation of a solute occur at equal rates in a saturated solution. In the case of AgBrO鈧, this equilibrium can be described using the solubility product constant ( K_{sp}).
The dissociation of AgBrO鈧 provides:
  • [Ag鈦篯 = 3.05 脳 10鈦烩伒 mol/L
  • [BrO鈧冣伝] = 3.05 脳 10鈦烩伒 mol/L
This is because the dissolution leads to equal concentrations of the resultant ions. The K_{sp} is calculated as the product of the ionic concentrations, K_{sp} = [Ag鈦篯[BrO鈧冣伝], resulting in K_{sp} = (3.05 脳 10鈦烩伒 mol/L) ^2 = 9.30 脳 10鈦宦光伆.
Understanding solubility equilibrium allows us to predict how substances dissolve, formulate chemical reactions, and solve diverse problems in chemistry.

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