Problem 71 - Chemistry An Atoms First Approach

Question

Chloral hydrate \(\left(\mathrm{C}_{2} \mathrm{H}_{3} \mathrm{Cl}_{3} \mathrm{O}_{2}\right)\) is a drug formerly used as a sedative and hypnotic. It is the compound used to make "Mickey Finns" in detective stories. a. Calculate the molar mass of chloral hydrate. b. What amount (moles) of \(\mathrm{C}_{2} \mathrm{H}_{3} \mathrm{Cl}_{3} \mathrm{O}_{2}\) molecules are in \(500.0 \mathrm{g}\) chloral hydrate? c. What is the mass in grams of \(2.0 \times 10^{-2}\) mole of chloral hydrate? d. What number of chlorine atoms are in 5.0 g chloral hydrate? e. What mass of chloral hydrate would contain 1.0 g Cl? f. What is the mass of exactly 500 molecules of chloral hydrate?

Step-by-Step Solution

Answer

The short answer for the problem is: a. The molar mass of chloral hydrate is \(165.9 \, \mathrm{g\, mol^{-1}}\). b. There are \(3.013 \, \mathrm{moles}\) of chloral hydrate in \(500.0 \mathrm{g}\). c. The mass of \(2.0 \times 10^{-2}\) mole of chloral hydrate is \(3.32 \, \mathrm{g}\). d. There are \(5.44 \times 10^{22}\) chlorine atoms in 5.0 g of chloral hydrate. e. A mass of \(1.56 \, \mathrm{g}\) of chloral hydrate contains 1.0 g of Cl. f. The mass of 500 molecules of chloral hydrate is \(1.38 \times 10^{-19} \, \mathrm{g}\).

Step 1: Identify the elements present and their atomic masses

Chloral hydrate has the formula \(\mathrm{C}_{2} \mathrm{H}_{3} \mathrm{Cl}_{3} \mathrm{O}_{2}\). It is made up of Carbon (C), Hydrogen (H), Chlorine (Cl), and Oxygen (O) atoms. We will use the following atomic masses: Carbon (C) = 12.01 g/mol Hydrogen (H) = 1.01 g/mol Chlorine (Cl) = 35.45 g/mol Oxygen (O) = 16.00 g/mol

Step 2: Calculate the molar mass of chloral hydrate

Now, we will calculate the molar mass of the chloral hydrate by multiplying the atomic masses of each element with their respective number of atoms in the formula and then summing up the results: Molar mass of chloral hydrate = \((2 \times 12.01) + (3 \times 1.01) + (3 \times 35.45) + (2 \times 16.00)\) Molar mass of chloral hydrate = \(165.9 \, \mathrm{g\, mol^{-1}}\) #b. What amount (moles) of \(\mathrm{C}_{2} \mathrm{H}_{3} \mathrm{Cl}_{3} \mathrm{O}_{2}\) molecules are in \(500.0 \mathrm{g}\) chloral hydrate?#

Step 1: Calculate the moles using the mass and molar mass

To find the number of moles in 500.0 g of chloral hydrate, we will use the formula: Moles = \(\frac{\mathrm{Mass}}{\mathrm{Molar\, Mass}}\) Substituting the values, we get Moles in 500.0 g chloral hydrate = \(\frac{500.0}{165.9}\) Moles = \(3.013 \, \mathrm{moles}\) #c. What is the mass in grams of \(2.0 \times 10^{-2}\) mole of chloral hydrate?#

Step 1: Calculate the mass using moles and molar mass

To find the mass in grams of \(2.0 \times 10^{-2}\) mole chloral hydrate, we will use the formula: Mass = Moles × Molar Mass Substituting the values, we get Mass = (\(2.0 \times 10^{-2}\) moles) × \(165.9 \, \mathrm{g/mol}\) Mass = \(3.32 \, \mathrm{g}\) #d. What number of chlorine atoms are in 5.0 g chloral hydrate?#

Step 1: Calculate the moles of chloral hydrate using mass and molar mass

First, we need to find the moles of chloral hydrate by using the given mass and calculated molar mass from part (a): Moles = \(\frac{5.0}{165.9}\) Moles = \(3.01 \times 10^{-2} \, \mathrm{moles}\)

Step 2: Calculate number of Chlorine atoms in 5.0 g chloral hydrate

Now we need to find the number of Chlorine atoms present in the moles of chloral hydrate. Three Chlorine atoms are present in each molecule of chloral hydrate. So, for moles of chloral hydrate, we will have three times the number of moles of Chlorine atoms. Moles of Chlorine atoms = \(3.01 \times 10^{-2} \times 3\) Now, we know that 1 mole has \(6.022 \times 10^{23}\) particles. So, to find the number of Chlorine atoms, multiply the moles of Chlorine atoms by Avogadro's number: Number of Chlorine atoms = \(3.01 \times 10^{-2} \times 3 \times 6.022 \times 10^{23}\) Number of Chlorine atoms = \(5.44 \times 10^{22}\) #e. What mass of chloral hydrate would contain 1.0 g Cl?#

Step 1: Calculate moles of Cl

First, find the moles of Cl using the given mass and atomic mass of Cl. Moles of Cl = \(\frac{1.0}{35.45}\) Moles of Cl = \(0.0282 \, \mathrm{moles}\)

Step 2: Calculate mass of chloral hydrate containing 1.0 g Cl

Since there are 3 moles of Cl in 1 mole of chloral hydrate, we can find the moles of chloral hydrate containing 0.0282 moles of Cl. Moles of chloral hydrate = \(\frac{0.0282}{3}\) Now, calculate the mass of chloral hydrate containing that many moles using its molar mass: Mass of chloral hydrate = (\(\frac{0.0282}{3}\) moles) × \(165.9 \, \mathrm{g/mol}\) Mass of chloral hydrate = \(1.56 \, \mathrm{g}\) #f. What is the mass of exactly 500 molecules of chloral hydrate?#

Step 1: Calculate moles for 500 molecules of chloral hydrate

First, convert the number of molecules to moles by dividing by Avogadro's number: Moles = \(\frac{500}{6.022 \times 10^{23}}\) Moles = \(8.3 \times 10^{-22} \, \mathrm{moles}\)

Step 2: Calculate mass using moles and molar mass

Now, calculate the mass of 500 molecules of chloral hydrate using the molar mass: Mass = (\(8.3 \times 10^{-22}\) moles) × \(165.9 \, \mathrm{g/mol}\) Mass = \(1.38 \times 10^{-19} \, \mathrm{g}\)

Key Concepts

Molar Mass Calculation

Understanding the molar mass of a compound is a fundamental concept in chemistry that often serves as a starting point for various calculations. The molar mass represents the mass of one mole of a substance, expressed in grams per mole (g/mol). To calculate the molar mass of chloral hydrate, a compound with the chemical formula \(\mathrm{C}_{2} \mathrm{H}_{3} \mathrm{Cl}_{3}\mathrm{O}_{2}\), we sum the masses of all the atoms present in one molecule.

Each atom has an atomic mass, which can be found on the periodic table: Carbon (C) is 12.01 g/mol, Hydrogen (H) is 1.01 g/mol, Chlorine (Cl) is 35.45 g/mol, and Oxygen (O) is 16.00 g/mol. By multiplying these atomic masses by their respective quantity in the formula and then adding them all together, we get the molar mass of chloral hydrate. For our example this calculation would be: \[ (2 \times 12.01) + (3 \times 1.01) + (3 \times 35.45) + (2 \times 16.00) = 165.9 \, \mathrm{g\, mol^{-1}}\].

Having the molar mass is crucial as it relates the mass of the substance to the number of moles, providing a path to subsequent stoichiometric calculations.

Converting Mass to Moles

To convert the mass of a substance to the number of moles, one simply divides the mass of the substance by its molar mass. This is effectively applying the formula \(\text{Moles} = \frac{\text{Mass}}{\text{Molar Mass}}\). In our problem involving chloral hydrate, we were asked to find out how many moles are in 500.0 g of the substance.

Using the molar mass we calculated earlier, 165.9 g/mol, we find the moles of chloral hydrate by dividing: \[\frac{500.0g}{165.9g/mol} = 3.013 \, \text{moles}\].

This step is instrumental for a multitude of applications in chemistry including reacting mass calculations, empirical and molecular formula determinations, and more importantly, it sets the stage for proper stoichiometric analysis.

Stoichiometry

Stoichiometry is the section of chemistry that deals with the quantitative relationships between the reactants and products in a chemical reaction. It operates under the premise that matter is conserved in these reactions, meaning the mass and number of atoms are the same before and after a chemical reaction.

One common stoichiometric problem might ask you to determine the mass of a reactant needed to produce a certain amount of product. For example, knowing the moles of chloral hydrate as calculated previously allows us to find out the mass corresponding to a given number of moles, using the inverse approach to the calculation in the second section. For instance, to find the mass in grams of \(2.0 \times 10^{-2}\) mole of chloral hydrate, the calculation would be: \[\text{Mass} = \text{Moles} \times \text{Molar Mass} = (2.0 \times 10^{-2} \text{ moles}) \times 165.9 \, \mathrm{g/mol} = 3.32 \, \mathrm{g}\].

This ability to interconvert between mass and moles is crucial for quantitative analysis in virtually all chemical processes.

Avogadro's Number Application

Avogadro's number, \(6.022 \times 10^{23}\), is the number of particles, such as atoms or molecules, in one mole of a substance. This fundamental constant enables chemists to count particles by weighing amounts of a substance. To determine the number of atoms or molecules in a given mass, we first convert the mass to moles and then use Avogadro’s number to convert those moles to number of particles.

For example, to calculate the number of chlorine atoms in 5.0 g of chloral hydrate, we first find the number of moles of chloral hydrate, and then we multiply that by 3, since there are 3 chlorine atoms per molecule, and by Avogadro's number to get the total atoms: \[\text{Number of Chlorine atoms} = \frac{5.0}{165.9} \times 3 \times 6.022 \times 10^{23} = 5.44 \times 10^{22}\].

The application of Avogadro's number allows us to manipulate and understand the world at the atomic and molecular level, providing a bridge between the macroscopic mass we measure and the microscopic particles that make up a substance.