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A 1.547 -g sample of blue copper(II) sulfate pentahydrate, \(\mathrm{CuSO}_{4} \cdot 5 \mathrm{H}_{2} \mathrm{O},\) is heated carefully to drive off the water. The white crystals of \(\mathrm{CuSO}_{4}\) that are left behind have a mass of \(0.989 \mathrm{~g}\). How many moles of \(\mathrm{H}_{2} \mathrm{O}\) were in the original sample? Show that the relative molar amounts of \(\mathrm{CuSO}_{4}\) and \(\mathrm{H}_{2} \mathrm{O}\) agree with the formula of the hydrate.

Short Answer

Expert verified
The original sample contained approximately 0.0310 moles of water, consistent with the hydrate formula CuSO4路5H2O.

Step by step solution

01

Determine Mass of Water Lost

The first step is to find the mass of the water lost during heating. It is obtained by subtracting the mass of anhydrous copper(II) sulfate (CuSO_{4}) from the original mass of the hydrate.\( \text{Mass of water} = \text{Mass of } \mathrm{CuSO}_{4} \cdot 5\mathrm{H}_{2} \mathrm{O} - \text{Mass of } \mathrm{CuSO}_{4}\)\(= 1.547 \, \text{g} - 0.989 \, \text{g} = 0.558 \, \text{g}\).
02

Calculate Moles of Water

Utilize the molar mass of water to determine moles of water lost. The molar mass of water (H_{2}O) is 18.015 g/mol. \( \text{Moles of } \mathrm{H}_{2}O = \frac{0.558 \, \text{g}}{18.015 \, \text{g/mol}} \approx 0.0310 \, \text{mol}\).
03

Calculate Moles of CuSO4

Now find the moles of anhydrous copper(II) sulfate (CuSO_{4}) using its molar mass (159.609 g/mol). \( \text{Moles of } \mathrm{CuSO}_{4} = \frac{0.989 \, \text{g}}{159.609 \, \text{g/mol}} \approx 0.00619 \, \text{mol}\).
04

Verify Hydrate Formula

To confirm the hydrate formula, compare the ratio of moles of water to moles of CuSO_{4}. Theoretically, the hydrate formula CuSO_{4} \cdot 5H_{2}O implies a 1:5 ratio.\( \text{Ratio} = \frac{0.0310 \, \text{mol H}_{2} \mathrm{O}}{0.00619 \, \text{mol CuSO}_{4}} \approx 5.01 \) This is very close to 5, supporting the given hydrate formula.

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Key Concepts

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

Molar Mass Calculation
When dealing with stoichiometry, a critical skill to master is calculating molar mass. Molar mass is the mass of one mole of a given substance and is usually expressed in grams per mole (g/mol). It serves as a bridge between the mass of a substance and the amount of substance, or moles, allowing us to make important calculations related to chemical reactions.

The molar mass of a compound is calculated by summing the atomic masses of its constituent elements, taken from the periodic table. Each element's atomic mass is usually noted as the average atomic mass of its isotopes. For a compound, multiply each element's atomic mass by the number of times the element appears in the formula, then add it up.
  • Example: For water (H鈧侽), calculate as follows: Hydrogen extbf{(H)} has an atomic mass of approximately 1.008 g/mol. Since water has two hydrogens, we use: \[ ext{Atomic mass of} ext{ H}_2 = 2 imes 1.008 ext{ g/mol} = 2.016 ext{ g/mol} \] Oxygen extbf{(O)} has an atomic mass of about 16.00 g/mol. \[ ext{Therefore, molar mass of H}_2 ext{O} = 2.016 ext{ g/mol} + 16.00 ext{ g/mol} = 18.015 ext{ g/mol} \]
Chemical Formulas
Chemical formulas are symbolic representations of chemical compounds. They reveal the elements present in a compound and the precise ratio of atoms in each element. Understanding and deciphering chemical formulas is critical for success in stoichiometry and chemical reactions.

There are different types of chemical formulas:
  • Empirical Formula: Shows the simplest integer ratio of atoms of each element in a compound. It may not show the actual number of atoms, but it gives a reduced proportion. For instance, the empirical formula of glucose (C鈧咹鈧佲倐O鈧) is CH鈧侽.
  • Molecular Formula: Gives the exact number of each type of atom in a molecule. In our example of glucose, the molecular formula is C鈧咹鈧佲倐O鈧, depicting the actual six carbons, twelve hydrogens, and six oxygens in a glucose molecule.

Understanding these formulas helps in calculating the stoichiometric relationships, predicting the product yields in reactions, and confirming stoichiometric coefficients in balanced equations.
Hydrates and Anhydrous Compounds
Hydrates are chemical compounds that include water molecules within their structure. The water molecules, known as 'water of crystallization,' are incorporated in specific stoichiometric proportions. A common example is copper(II) sulfate pentahydrate ( CuSO鈧劼5H鈧侽 ), which incorporates five water molecules for each copper sulfate unit.

When hydrates are heated, the water of crystallization is expelled, leaving behind an anhydrous compound. The mass change due to water loss is used to determine the amount of water present in the original hydrate. Detection and quantifying these change is crucial in empirical and industrial chemistry.
  • Anhydrous Compound: A compound that has had its water molecules removed, typically by heating. An example is anhydrous copper(II) sulfate (CuSO鈧), which is the form left after heating copper(II) sulfate pentahydrate.
  • Practical Application: Understanding and identifying hydrates versus anhydrous compounds is important for quality control in various chemical processes.

In our exercise, by heating CuSO鈧劼5H鈧侽, we drive off the water, confirming the number of water molecules initially present through stoichiometric calculations. This helps verify the compound鈥檚 correctness and ensure it aligns with theoretical expectations.

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Most popular questions from this chapter

A friend is doing his chemistry homework and is working with the following chemical reaction. $$ 2 \mathrm{C}_{2} \mathrm{H}_{2}(g)+5 \mathrm{O}_{2}(g) \longrightarrow 4 \mathrm{CO}_{2}(g)+2 \mathrm{H}_{2} \mathrm{O}(g) $$ He tells you that if he reacts 2 moles of \(\mathrm{C}_{2} \mathrm{H}_{2}\) with 4 moles of \(\mathrm{O}_{2}\), then the \(\mathrm{C}_{2} \mathrm{H}_{2}\) is the limiting reactant since there are fewer moles of \(\mathrm{C}_{2} \mathrm{H}_{2}\) than \(\mathrm{O}_{2}\). How would you explain to him where he went wrong with his reasoning (what concept is he missing)? After providing your friend with the explanation from part a, he still doesn't believe you because he had a homework problem where 2 moles of calcium were reacted with 4 moles of sulfur and he needed to determine the limiting reactant. The reaction is $$ \mathrm{Ca}(s)+\mathrm{S}(s) \longrightarrow \operatorname{CaS}(s) $$ He obtained the correct answer, Ca, by reasoning that since there were fewer moles of calcium reacting, calcium had to be the limiting reactant. How would you explain his reasoning flaw and why he got "lucky" in choosing the answer that he did?

Carbon disulfide, \(\mathrm{CS}_{2}\), is a colorless, highly flammable liquid used in the manufacture of rayon and cellophane. A sample contains \(0.0116 \mathrm{~mol} \mathrm{CS}_{2}\). Calculate the mass of carbon disulfide in the sample.

Nitric acid, \(\mathrm{HNO}_{3}\), is manufactured by the Ostwald process, in which nitrogen dioxide, \(\mathrm{NO}_{2}\), reacts with water. $$ 3 \mathrm{NO}_{2}(g)+\mathrm{H}_{2} \mathrm{O}(l) \longrightarrow 2 \mathrm{HNO}_{3}(a q)+\mathrm{NO}(g) $$ How many grams of nitrogen dioxide are required in this reaction to produce \(8.60 \mathrm{~g} \mathrm{HNO}_{3} ?\)

A moth repellent, para-dichlorobenzene, has the composition \(49.1 \% \mathrm{C}, 2.7 \% \mathrm{H},\) and \(48.2 \% \mathrm{Cl} .\) Its molecular weight is 147 amu. What is its molecular formula?

Thiophene is a liquid compound of the elements C, \(\mathrm{H},\) and \(\mathrm{S} .\) A sample of thiophene weighing \(7.96 \mathrm{mg}\) was burned in oxygen, giving \(16.65 \mathrm{mg} \mathrm{CO}_{2}\). Another sample was subjected to a series of reactions that transformed all of the sulfur in the compound to barium sulfate. If \(4.31 \mathrm{mg}\) of thiophene gave \(11.96 \mathrm{mg}\) of barium sulfate, what is the empirical formula of thiophene? Its molecular weight is 84 amu. What is its molecular formula?

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