Question

Vitamin \(\mathrm{B}_{12}\), cyanocobalamin, is essential for human nutrition. Its molecular formula is \(\mathrm{C}_{63} \mathrm{H}_{88} \mathrm{CoN}_{14} \mathrm{O}_{14} \mathrm{P}\). A lack of this vitamin in the diet can lead to anemia. Cyanocobalamin is the form of the vitamin found in vitamin supplements. a. What is the molar mass of cyanocobalamin to two decimal places? b. How many moles of cyanocobalamin molecules are present in \(250 \mathrm{mg}\) of cyanocobalamin? c. What is the mass of 0.60 mole of cyanocobalamin? d. How many atoms of hydrogen are in 1.0 mole of cyanocobalamin? e. What is the mass of \(1.0 \times 10^{7}\) molecules of cyanocobalamin? f. What is the mass (in grams) of one molecule of cyanocobalamin?

Short answer

a. The molar mass of cyanocobalamin is 1355.39 g/mol. b. There are 1.84 x 10^-4 moles of cyanocobalamin in 250 mg. c. The mass of 0.60 moles of cyanocobalamin is 813.24 g. d. There are 5.29 x 10^25 hydrogen atoms in 1.0 mole of cyanocobalamin. e. The mass of \(1.0 \times 10^{7}\) molecules of cyanocobalamin is 2.25 x 10^-14 g. f. The mass of one molecule of cyanocobalamin is 2.25 x 10^-21 g.

Step-by-step solution

Identify the elements and their atomic masses from the molecular formula

For cyanocobalamin, the molecular formula is \(\mathrm{C}_{63} \mathrm{H}_{88} \mathrm{CoN}_{14} \mathrm{O}_{14} \mathrm{P}\). The atomic masses we'll need are: - Carbon (C): 12.01 g/mol - Hydrogen (H): 1.008 g/mol - Cobalt (Co): 58.93 g/mol - Nitrogen (N): 14.01 g/mol - Oxygen (O): 16.00 g/mol - Phosphorus (P): 30.97 g/mol

Multiply each atomic mass by the number of atoms in the molecular formula

Using the molecular formula and the atomic masses, multiply each atomic mass by the number of atoms for each element present in the molecule: - C: 63 * 12.01 g/mol - H: 88 * 1.008 g/mol - Co: 1 * 58.93 g/mol - N: 14 * 14.01 g/mol - O: 14 * 16.00 g/mol - P: 1 * 30.97 g/mol

Add them all together to find the molar mass

Now add up all the values obtained in Step 2 to find the molar mass of cyanocobalamin: Molar Mass = (63 * 12.01) + (88 * 1.008) + 58.93 + (14 * 14.01) + (14 * 16.00) + 30.97 Molar Mass (to two decimal places) = 1355.39 g/mol #b. Calculate moles from mass#

Convert mass to grams

Convert the given mass of cyanocobalamin, 250 mg, to grams: 250 mg (1 g / 1000 mg) = 0.25 g

Divide the mass by the molar mass

Divide the mass of cyanocobalamin (0.25 g) by its molar mass (1355.39 g/mol) to find the number of moles: Moles = 0.25 g / 1355.39 g/mol = 1.84 x 10^-4 mol #c. Calculate the mass of 0.60 moles of cyanocobalamin#

Multiply the molar mass by the number of moles

Find the mass of 0.60 moles of cyanocobalamin by multiplying its molar mass (1355.39 g/mol) by the number of moles (0.60 moles): Mass = 1355.39 g/mol * 0.60 mol = 813.24 g #d. Calculate the number of hydrogen atoms in 1 mole of cyanocobalamin#

Identify the number of hydrogen atoms in the molecular formula

In cyanocobalamin, there are 88 hydrogen atoms (from the molecular formula \(\mathrm{C}_{63} \mathrm{H}_{88} \mathrm{CoN}_{14} \mathrm{O}_{14} \mathrm{P}\)).

Multiply by Avogadro's Number

Multiply the number of hydrogen atoms per molecule (88 H) by Avogadro's Number (6.022 x 10^23 molecules/mol): Hydrogen atoms = 88 H * 6.022 x 10^23 molecules/mol = 5.29 x 10^25 hydrogen atoms #e. Calculate the mass of \(1.0 \times 10^{7}\) molecules of cyanocobalamin#

Calculate moles from the number of molecules

To find the moles of cyanocobalamin, divide the number of molecules (1.0 x 10^7) by Avogadro's Number (6.022 x 10^23): Moles = 1.0 x 10^7 molecules / 6.022 x 10^23 molecules/mol = 1.66 x 10^-17 mol

Multiply moles by the molar mass

Multiply the moles of cyanocobalamin (1.66 x 10^-17 mol) by its molar mass (1355.39 g/mol) to find the mass: Mass = 1.66 x 10^-17 mol * 1355.39 g/mol = 2.25 x 10^-14 g #f. Calculate the mass of one molecule of cyanocobalamin#

Calculate moles of one molecule

Since one molecule is equivalent to 1/Avogadro's Number moles, we have: Moles = 1/6.022 x 10^23 = 1.66 x 10^-24 mol

Multiply moles by the molar mass

Multiply the moles of one cyanocobalamin molecule (1.66 x 10^-24 mol) by its molar mass (1355.39 g/mol) to find the mass of one molecule: Mass = 1.66 x 10^-24 mol * 1355.39 g/mol = 2.25 x 10^-21 g

Key concepts

Molar Mass Calculation

To determine the molar mass of a complex molecule like cyanocobalamin, you start by examining its molecular formula: \( \mathrm{C}_{63} \mathrm{H}_{88} \mathrm{CoN}_{14} \mathrm{O}_{14} \mathrm{P} \). This formula tells us how many atoms of each element are present. The steps involved are:

• List the elements and their quantities in the compound.
• Obtain the atomic mass of each element from a periodic table; for example, carbon (C) is 12.01 g/mol.
• Multiply the atomic mass by the number of each type of atom: for 63 atoms of carbon, it's \( 63 \times 12.01 \), resulting in a contribution of 756.63 g/mol to the molar mass.
• Repeat this for all elements: hydrogen (H), cobalt (Co), nitrogen (N), oxygen (O), and phosphorus (P).
• Add the results: the sum gives the total molar mass of the molecule.

For cyanocobalamin, this total adds up to 1355.39 g/mol. Understanding molar mass is crucial as it serves as a conversion factor for determining the amounts of substances involved in a reaction.

Avogadro's Number

Avogadro's number is a fundamental concept in chemistry that defines the number of atoms, molecules, or particles in one mole of a substance. It is approximately \( 6.022 \times 10^{23} \). This large constant bridges the microscopic world of atoms to macroscopic measurements like grams or liters. Here's how it's utilized:

• It enables conversion between individual molecules and their corresponding mole amounts. For example, knowing the moles of a substance allows you to calculate the number of molecules by multiplying the moles by \( 6.022 \times 10^{23} \).
• In calculations involving molecules, like those with cyanocobalamin, it simplifies finding the number of specific atoms, such as hydrogen. If you have one mole, multiply Avogadro's number by the quantity of that atom per molecule (88 hydrogens in this case) to obtain the total atoms present.

By using Avogadro's number, chemists can accurately work with the incredibly high numbers of atoms while translating them into measurable quantities.

Conversion Between Moles and Grams

One key skill in chemistry is converting between moles and grams, allowing us to work seamlessly with a substance's mass and its chemical amount. Grams and moles are connected through the molar mass, as explained above. Here's the conversion process:

• To convert grams of a substance to moles, divide the mass by the substance's molar mass. For example, for cyanocobalamin: \( \frac{0.25\, \text{g}}{1355.39\, \text{g/mol}} = 1.84 \times 10^{-4} \text{ moles} \).
• Conversely, to find out how much a given number of moles weighs, multiply the moles by the molar mass. For 0.60 moles of cyanocobalamin, you calculate \( 0.60 \times 1355.39 = 813.24 \text{ g} \).

These conversions are fundamental, as they enable one to translate lab measurements into chemical equations and vice versa, ensuring stoichiometrically accurate chemical reactions.