Problem 35 - Chemistry: The Central Science

Question

Calculate the following quantities: (a) mass, in grams, of \(0.105 \mathrm{~mol}\) sucrose \(\left(\mathrm{C}_{12} \mathrm{H}_{22} \mathrm{O}_{11}\right)\) (b) moles of \(\mathrm{Zn}\left(\mathrm{NO}_{3}\right)_{2}\) in \(143.50 \mathrm{~g}\) of this substance (c) number of molecules in \(1.0 \times 10^{-6} \mathrm{~mol} \mathrm{} \mathrm{CH}_{3} \mathrm{CH}_{2} \mathrm{OH}\) (d) number of \(\mathrm{N}\) atoms in \(0.410 \mathrm{~mol} \mathrm{} \mathrm{NH}_{3}\)

Step-by-Step Solution

Answer

(a) Mass of sucrose = \(0.105 \mathrm{~mol} \times 342.30 \mathrm{~g/mol} = 35.9415 \mathrm{~g}\) (b) Moles of Zn(NO3)₂ = \(143.50 \mathrm{~g} \div 189.42 \mathrm{~g/mol} = 0.7574 \mathrm{~mol}\) (c) Number of molecules in CH3CH2OH = \(1.0 \times 10^{-6} \mathrm{~mol} \times 6.022 \times 10^{23} \mathrm{~molecules/mol} = 6.022 \times 10^{17} \mathrm{~molecules}\) (d) Number of N atoms in NH3 = \(0.410 \mathrm{~mol} \times 6.022 \times 10^{23} \mathrm{~atoms/mol}\times 1 = 2.469 \times 10^{23} \mathrm{~N~atoms}\)

Step 1: Calculate the molar mass of sucrose

To calculate the molar mass of sucrose (C12H22O11), we need to add the molar masses of each element in the formula, multiplying by the number of each element: Molar mass of C = 12.01 g/mol Molar mass of H = 1.01 g/mol Molar mass of O = 16.00 g/mol Molar mass of sucrose = 12 * Molar mass of C + 22 * Molar mass of H + 11 * Molar mass of O

Step 2: Calculate the mass of sucrose

Now, multiply the moles of sucrose by its molar mass to find the mass in grams: Mass of sucrose = Moles of sucrose * Molar mass of sucrose ------ #b) Moles of Zn(NO3)2#

Step 1: Calculate the molar mass of Zn(NO3)2

To calculate the molar mass of Zn(NO3)2, we need to add the molar masses of each element in the formula, multiplying by the number of each element: Molar mass of Zn = 65.38 g/mol Molar mass of N = 14.01 g/mol Molar mass of O = 16.00 g/mol Molar mass of Zn(NO3)₂ = Molar mass of Zn + 2 * (Molar mass of N + 3 * Molar mass of O)

Step 2: Calculate the moles of Zn(NO3)₂

Now, divide the mass of Zn(NO3)₂ by its molar mass to find the moles: Moles of Zn(NO3)₂ = Mass of Zn(NO3)₂ / Molar mass of Zn(NO3)₂ ------ #c) Number of molecules in CH3CH2OH#

Step 1: Determine Avogadro's number

Avogadro's number is the number of atoms, ions, or molecules in one mole of any substance. It is equal to \(6.022 \times 10^{23}\) particles/mole.

Step 2: Calculate the number of molecules

Now, multiply the moles of CH3CH2OH by Avogadro's number to find the number of molecules: Number of molecules = Moles of CH3CH2OH * Avogadro's number ------ #d) Number of N atoms in NH3#

Step 1: Determine the number of N atoms in one molecule of NH3

In one molecule of NH3, there is one N atom.

Step 2: Calculate the number of N atoms

Now, multiply the moles of NH3 by Avogadro's number and the number of N atoms in one molecule of NH3 to find the total number of N atoms: Number of N atoms = Moles of NH3 * Avogadro's number * Number of N atoms in one NH3 molecule

Key Concepts

Molar Mass Calculation

Understanding the molar mass of a substance is a fundamental aspect of stoichiometry calculations, which are essential for understanding chemical reactions and creating balanced equations. The molar mass is the weight of one mole of a given substance, typically expressed in grams per mole (g/mol).

To calculate the molar mass, one must sum the atomic masses of all the atoms in a molecule. Atomic masses can be found on the periodic table and are usually averaged due to the existence of isotopes. Let's take sucrose (C₁₂H₂₂O₁₁) as an example. We multiply the atomic mass of carbon (C) by 12, the atomic mass of hydrogen (H) by 22, and the atomic mass of oxygen (O) by 11, and then sum these values to obtain the molar mass of sucrose.

Avogadro's Number

A cornerstone of chemistry is Avogadro's number, which is used in mole-to-particle conversions. This constant, defined as approximately 6.022 x 10²³, represents the number of atoms, ions or molecules in one mole of substance. It's named after Amedeo Avogadro, an Italian scientist who contributed to molecular theory.

Avogadro's number lets us connect the microscopic world of atoms and molecules to the macroscopic world we can measure and observe. For instance, when converting moles to the number of molecules, we multiply the number of moles by Avogadro's number, as seen in the conversion for ethanol (CH₃CH₂OH).

Mole-to-Mass Conversion

Mole-to-mass conversion involves translating moles, a unit for the amount of substance, into grams, a unit for mass. To convert moles to mass, we use the equation:
Mass (g) = Moles (mol) x Molar Mass (g/mol).

This calculation is crucial when measuring out reactive agents in a laboratory or understanding how much of a chemical is present in a given sample. For example, after calculating the molar mass of sucrose, we use it to convert moles of sucrose to grams, allowing for precise measurement of this sweet compound.

Mole-to-Particle Conversion

Closely related to Avogadro's number, mole-to-particle conversion enables chemists to count the exact number of discrete entities (atoms, molecules, ions) in a given amount of substance. The formula to do this conversion is straightforward:
Number of particles = Moles (mol) x Avogadro's number (particles/mol).

For instance, in the exercise, we determined the number of nitrogen (N) atoms in ammonia (NH₃) by multiplying the amount in moles by Avogadro’s number. This conversion provides a clear bridge from the quantitative moles to the actual particles involved in chemical reactions.