Question

A 0.755-g sample of hydrated copper(II) sulfate $$\mathrm{CuSO}_{4} \cdot x \mathrm{H}_{2} \mathrm{O}$$ was heated carefully until it had changed completely to anhydrous copper(II) sulfate (CuSO_) with a mass of 0.483 g. Determine the value of \(x .\) [This number is called the number of waters of hydration of copper(II) sulfate. It specifies the number of water molecules per formula unit of \(\mathrm{CuSO}_{4}\) in the hydrated crystal.]

Short answer

The hydrated copper(II) sulfate formula is CuSO\(_4 \cdot 5\)H\(_2\)O, where the value of \(x\) is 5.

Step-by-step solution

Calculate the mass of water evaporated during heating

Subtract the mass of anhydrous copper(II) sulfate from the mass of hydrated copper(II) sulfate to find the mass of water evaporated: mass of water evaporated = mass of hydrated copper(II) sulfate - mass of anhydrous copper(II) sulfate mass of water evaporated = 0.755 g - 0.483 g = 0.272 g

Calculate the molar mass of anhydrous copper(II) sulfate (CuSO\(_4\)) and water (H\(_2\)O)

To find the value of \(x\), we need to know the molar mass of the substances involved. Use the periodic table to find the molar mass of each element and then add them up accordingly: Molar mass of CuSO\(_4\) = 63.55 (Cu) + 32.07 (S) + 4 * 16.00 (O) = 159.62 g/mol Molar mass of H\(_2\)O = 2 * 1.01 (H) + 16.00 (O) = 18.02 g/mol

Calculate the moles of CuSO\(_4\) and water (H\(_2\)O) involved

Divide the masses of each substance by their respective molar mass to find the number of moles: moles of CuSO\(_4\) = mass of CuSO\(_4\) / molar mass of CuSO\(_4\) = 0.483 g / 159.62 g/mol ≈ 0.00303 mol moles of H\(_2\)O = mass of H\(_2\)O / molar mass of H\(_2\)O = 0.272 g / 18.02 g/mol ≈ 0.0151 mol

Determine the value of \(x\) considering the ratio of moles of CuSO\(_4\) and H\(_2\)O

Divide the moles of H\(_2\)O by the moles of CuSO\(_4\) to find the value of \(x\): x = moles of H\(_2\)O / moles of CuSO\(_4\) = 0.0151 mol / 0.00303 mol ≈ 5 Since \(x\) must be an integer, we can conclude that \(x = 5\). Hence, the hydrated copper(II) sulfate formula should be written as: CuSO\(_4 \cdot 5\)H\(_2\)O

Key concepts

Copper(II) sulfate

Copper(II) sulfate is a versatile chemical compound with a wide range of applications. It's commonly found in its pentahydrate form, which means it has a formula of \(\text{CuSO}_4 \cdot 5\text{H}_2\text{O}\). This distinctive crystal form is blue and is often used in experiments due to its clear and striking color changes when heated.
When heated, hydrated copper(II) sulfate loses its water of hydration and turns into an anhydrous form. This anhydrous copper(II) sulfate is white or off-white, marking a visual transition from the original blue compound.
This property is exploited in laboratories to explore the concept of hydration and dehydration, providing a vivid demonstration of physical changes due to the loss of water molecules.

Anhydrous compound

An anhydrous compound is one that contains no water molecules in its structure. It's the dry form of a chemical that, in its hydrated form, is associated with water molecules. In the exercise, anhydrous copper(II) sulfate is produced by heating its hydrated counterpart to remove all water molecules.
The physical difference is notable; while the hydrated form is vividly colored due to water molecules interfering with the copper ions, the anhydrous form is pale, as the absence of water changes the crystal structure.
Understanding anhydrous compounds is essential in chemistry as they are often used in desiccants and other applications where water presence needs control to avoid interference with processes.

Molar mass calculation

Molar mass calculation is a fundamental skill in chemistry, allowing us to convert between mass and moles of a substance. It involves using the periodic table to find the atomic masses of elements in a compound and summing these values to get the compound's molar mass.
In the case of copper(II) sulfate, we calculate the molar mass of \(\text{CuSO}_4\) by adding the atomic masses of copper (63.55 g/mol), sulfur (32.07 g/mol), and oxygen, where four oxygen atoms contribute a total of 64.00 g/mol. This results in a total molar mass of 159.62 g/mol for anhydrous copper(II) sulfate.
Similarly, water has a molar mass of 18.02 g/mol, calculated by adding two hydrogen atoms and one oxygen atom. These calculations are used to find the number of moles of each substance and, ultimately, to determine the number of water molecules in the hydrated form of copper(II) sulfate.