The Ksp of CaSO4 is 4.93× 10–5. Calculate the solubility (in g/L) of CaSO4(s) in 0.300 M Na2SO4(aq) at 25 °C

To solve this problem, we need to use the concept of common-ion effect. The common-ion effect describes the phenomenon where the solubility of a slightly soluble salt is reduced by the presence of a common ion (from a soluble source).

The balanced equation for the dissolution of calcium sulfate (CaSO4) is:

CaSO4(s) ⇌ Ca2+(aq) + SO4^2-(aq)

Since we already have the Ksp value, we can write the expression for the Ksp as follows:

Ksp = [Ca2+][SO4^2-]

We need to consider that when calcium sulfate dissolves in 0.300 M sodium sulfate, it will produce additional sulfate ions. Therefore, the concentration of sulfate ions in the solution will be the sum of the initial concentration from sodium sulfate (0.300 M) and the concentration produced through the dissolution of calcium sulfate.

Let's assume that x moles of calcium sulfate dissolve.

The concentrations of Ca2+ and SO4^2- ions will be:

[Ca2+] = x
[SO4^2-] = 0.300 + x

Substituting these values into the Ksp expression, we get:

Ksp = (x)(0.300 + x)

Now, we can calculate the solubility of CaSO4 by solving this equation for x.

Ksp = 4.93 × 10^(-5)
0.300 + x = x

(x)(0.300 + x) = 4.93 × 10^(-5)

x^2 + 0.300x - 4.93 × 10^(-5) = 0

This is a quadratic equation in terms of x, and it can be solved using the quadratic formula.

To calculate the solubility of CaSO4 in 0.300 M Na2SO4, we need to consider the common ion effect. The presence of Na2SO4 in the solution will provide additional SO4^2- ions, which will suppress the solubility of CaSO4.

To find the solubility of CaSO4, we first need to write the balanced chemical equation for the dissolution of CaSO4 in water:

CaSO4 (s) ⇌ Ca^2+ (aq) + SO4^2- (aq)

The solubility product expression (Ksp) for this reaction is:

Ksp = [Ca^2+] × [SO4^2-]

Given that the Ksp of CaSO4 is 4.93 × 10^–5, we can assume that the concentration of Ca^2+ and SO4^2- ions in the saturated solution are equal.

Let's assume the solubility of CaSO4 in 0.300 M Na2SO4 is "x" moles per liter. The concentration of SO4^2- ions in the solution will be the sum of the SO4^2- ions coming from the Na2SO4 and the dissolved CaSO4. Since 1 mole of CaSO4 produces 1 mole of SO4^2- ions, we can write the expression for the concentration of SO4^2- ions as:

[SO4^2-] = 0.300 M + x

Since the concentration of Ca^2+ ions is equal to the concentration of SO4^2- ions, [Ca^2+] = [SO4^2-] = 0.300 M + x.

Now we can substitute these values into the solubility product expression:

Ksp = (0.300 M + x) × (0.300 M + x)

Since we know the Ksp value is 4.93 × 10^–5, we can solve for "x":

4.93 × 10^–5 = (0.300 + x)(0.300 + x)

Expanding the equation:

4.93 × 10^–5 = 0.09 + 0.6x + x^2

Rearranging the equation:

x^2 + 0.6x + 0.09 - 4.93 × 10^–5 = 0

Now solve this quadratic equation using either factoring, completing the square, or the quadratic formula. In this case, it is simpler to use the quadratic formula:

x = (-0.6 ± √((-0.6)^2 - 4(1)(0.09 - 4.93 × 10^–5))) / 2(1)

Simplifying:

x = (-0.6 ± √(0.36 - 4.93 × 10^–5)) / 2

Calculating the discriminant:

x = (-0.6 ± √(0.3599999757)) / 2

x ≈ (-0.6 ± 0.6) / 2

x ≈ 0 or x ≈ 0.3

Since we are interested in the solubility of CaSO4 in g/L, we need to convert "x" from moles per liter to grams per liter. The molar mass of CaSO4 is approximately 136.14 g/mol.

Using x = 0.3 moles/L:

Solubility = x × molar mass = 0.3 mol/L × 136.14 g/mol = 40.84 g/L

Therefore, the solubility (in g/L) of CaSO4(s) in 0.300 M Na2SO4(aq) at 25 °C is approximately 40.84 g/L.

........CaSO4 ==> Ca^2+ + SO4^2-

I........solid.....0........0
C........solid.....x........x
E........solid.....x........x

For Na2SO4(aq) ==> 2Na^+ + SO4^2-
I.....0.300M........0........0
C......-0.300......0.300*2...0.300
E........0.........0.600....0.300

Ksp CaSO4 = (Ca^2+)(SO4^2-)
For Ca you can see that is x as well as the solubility.
For SO4^2- you have two sources. From CaSO4 it is x; from Na2SO4 it is 0.300; the total is 0.300+x

Substitute into Ksp expression and solve for x = molar solubility CaSO4.
For