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 \(\left(\mathrm{CuSO}_{4}\right)\) with a mass of 0.483 g. Determine the value of \(x .[\text { This number is called the number of waters }\) of hydration of copper(Il) sulfate. It specifies the number of water molecules per formula unit of \(\mathrm{CuSO}_{4}\) in the hydrated crystal. \(]\)

Short answer

The value of x, which represents the number of water molecules per formula unit of CuSO4 in the hydrated crystal, is approximately 5. Therefore, the formula of the hydrated copper(II) sulfate is given by: CuSO4 · 5H2O.

Step-by-step solution

1. Calculate the mass of water lost during the heating process

To find the mass of water lost during the heating process, subtract the mass of the anhydrous copper(II) sulfate from the mass of the hydrated copper(II) sulfate: Mass of water = Initial mass of hydrated copper(II) sulfate - Final mass of anhydrous copper(II) sulfate Mass of water = \(0.755 \, g - 0.483 \, g\) Mass of water = \(0.272 \, g\)

2. Determine the moles of anhydrous copper(II) sulfate and the moles of water

Next, we need to calculate the moles of anhydrous copper(II) sulfate and the moles of water. To do this, divide their respective masses by their molar masses. The molar mass of copper(II) sulfate (CuSO4) is approximately \(63.5 + 32 + (4 \times 16) \, g/mol = 159.5 \, g/mol\), and the molar mass of water (H2O) is approximately \(2 + 16 = 18 \, g/mol\). Moles of anhydrous copper(II) sulfate: Moles of CuSO4 = \( \frac{0.483 \, g}{159.5 \, g/mol} \) Moles of CuSO4 = \(0.00303 \, mol\) Moles of water: Moles of H2O = \( \frac{0.272 \, g}{18 \, g/mol} \) Moles of H2O = \( 0.0151 \, mol\)

3. Calculate the ratio between moles of water and moles of CuSO4

To find the value of x, we need to calculate the ratio between the moles of water and moles of anhydrous copper(II) sulfate: x = Moles of H2O / Moles of CuSO4 x = \( \frac{0.0151 \, mol}{0.00303 \, mol} \) x ≈ 5

Conclusion

The value of x, which represents the number of water molecules per formula unit of CuSO4 in the hydrated crystal, is approximately 5. Therefore, the formula of the hydrated copper(II) sulfate is given by: CuSO4 · 5H2O

Key concepts

Molar Mass Calculation

In chemistry, calculating molar mass is a fundamental step in understanding the composition of molecules. Each element in a compound contributes to the total mass, and molar mass provides a way to quantify this. For a hydrated compound like copper(II) sulfate hydrate, you need to consider both copper(II) sulfate and the associated water molecules.

For copper(II) sulfate (\(\mathrm{CuSO}_{4}\)), you sum up the atomic masses based on periodic table values:

• Copper (Cu): approximately 63.5 g/mol
• Sulfur (S): 32.0 g/mol
• Oxygen (O): 16.0 g/mol per atom, with four atoms contributing a total of 64.0 g/mol

When you add these up, the molar mass of copper(II) sulfate is \(63.5 + 32 + 64 = 159.5\) g/mol.

Similarly, for water (\(\mathrm{H}_{2}\mathrm{O}\)), the calculation is straightforward:

• Hydrogen (H): 1.0 g/mol per atom, contributing a total of 2.0 g/mol for two hydrogen atoms
• Oxygen (O): 16.0 g/mol

This gives the molar mass of water as \(2 + 16 = 18\) g/mol.

Moles and Molecules

When solving chemistry problems, understanding moles and how they relate to molecules is crucial. A mole is a unit in chemistry that provides a bridge between the atomic scale and the macroscopic world. It represents \(6.022 \times 10^{23}\) entities, which could be atoms, ions, or molecules, known as Avogadro's number.

In our exercise, to find out how many moles of water and copper(II) sulfate are present, we divide their masses by their respective molar masses.

For anhydrous copper(II) sulfate:

• Mass given is 0.483 g
• Molar mass is 159.5 g/mol
• Moles of \(\mathrm{CuSO}_{4} = \frac{0.483\, \text{g}}{159.5\, \text{g/mol}}\)
• Moles of \(\mathrm{CuSO}_{4} \approx 0.00303\, \text{mol}\)

For water:

• Mass lost is 0.272 g
• Molar mass is 18 g/mol
• Moles of \(\mathrm{H}_{2} \mathrm{O} = \frac{0.272\, \text{g}}{18\, \text{g/mol}}\)
• Moles of \(\mathrm{H}_{2} \mathrm{O} \approx 0.0151\, \text{mol}\)

Using moles helps us measure large quantities of tiny particles, enabling accurate stoichiometric calculations.

Chemistry Problem Solving

Solving chemistry problems involves a systematic approach. Always begin by analyzing the data provided and the relationships between components. In the problem with hydrated copper(II) sulfate, steps include calculating the change in mass due to heating, identifying the moles involved, and ultimately finding the number of hydrates, \(x\).

First, determine the mass of the lost water by subtracting the final mass (anhydrous) from the initial mass (hydrous). Next, calculate the moles for both substances using their molar masses. The objective is to find the ratio of water to copper(II) sulfate, represented as \(x\).

This ratio is found by dividing the moles of water by the moles of anhydrous compound:

• \(x = \frac{\text{Moles of } \mathrm{H}_{2} \mathrm{O}}{\text{Moles of } \mathrm{CuSO}_{4}}\)
• With numbers, \(x = \frac{0.0151\, \text{mol}}{0.00303\, \text{mol}} \approx 5\)

The final calculation shows there are approximately five water molecules per copper(II) sulfate unit, confirming the formula \(\mathrm{CuSO}_{4} \cdot 5\mathrm{H}_{2} \mathrm{O}\). By practicing this structured problem-solving method, you build confidence in tackling various chemistry challenges.