Question

The formula for vitamin \(\mathrm{B}_{12}\) is \(\mathrm{C}_{63} \mathrm{H}_{88} \mathrm{CoN}_{14} \mathrm{O}_{14} \mathrm{P}\). (a) How many atoms make up one molecule of vitamin \(\mathrm{B}_{12}\) ? (b) What percentage of the total atoms are carbon? (c) What fraction of the total atoms are metallic?

Short answer

(a) 181 atoms (b) 34.81% carbon (c) \(\frac{1}{181}\) metallic

Step-by-step solution

Count the Total Number of Atoms

Add up the number of each type of atom in the formula. \(63\) (carbon) \(+ 88\) (hydrogen) \(+ 1\) (cobalt) \(+ 14\) (nitrogen) \(+ 14\) (oxygen) \(+ 1\) (phosphorus) = \(181\) atoms.

Calculate the Percentage of Carbon Atoms

Divide the number of carbon atoms by the total number of atoms and multiply by 100: \(\left( \frac{63}{181} \right) \times 100 \approx 34.81\%\).

Determine the Fraction of Metallic Atoms

Here, only the cobalt atom is metallic. The fraction is given by: \(\frac{1}{181}\).

Key concepts

headline of the respective core concept

To get a grip on molecular composition problems, understanding chemical formula analysis is essential. Let's break down the chemical formula of vitamin \(\text{B}_{12}\). The formula \(\text{C}_{63} \text{H}_{88} \text{CoN}_{14} \text{O}_{14} \text{P}\) indicates the number of each type of atom in one molecule

In these types of problems, your first step should be to count the atoms. By adding up all the subscript numbers: 63 (from C) + 88 (from H) + 1 (from Co) + 14 (from N) + 14 (from O) + 1 (from P), you'll see there are a total of 181 atoms in one molecule. This count helps us understand the detailed structure of the compound.

Breaking down the formula like this is crucial for subsequent calculations, such as determining the percentage of different elements and calculating molecular fractions.

headline of the respective core concept

One common question is how to find the percentage composition of a compound. For vitamin \(\text{B}_{12}\), if we want to find the percentage of carbon atoms, we follow a simple formula:

Divide the number of carbon atoms (which is 63) by the total number of atoms (which is 181): \(\frac{63}{181}\).

To convert this fraction to a percentage, multiply by 100: \( \frac{63}{181} \times 100 \approx 34.81\%\)

This means that about 34.81% of the atoms in one molecule of vitamin \(\text{B}_{12}\) are carbon atoms.

Understanding how to calculate percentages like this is a fundamental part of chemical formula analysis. Being able to express the amount of a specific atom in a compound as a percentage is useful in many chemical and biological applications.

headline of the respective core concept

Another interesting aspect is determining molecular fractions. In the case of vitamin \(\text{B}_{12}\), we can find the fraction of metallic atoms.

We know that cobalt is the only metallic atom in the formula. There's just one cobalt atom in the total of 181 atoms. To find the molecular fraction of cobalt, divide 1 by 181: \(\frac{1}{181}\).

This fraction tells us that metallic atoms make up a very small part of the total number of atoms in a molecule of vitamin \(\text{B}_{12}\).

Whether dealing with vitamins or any other substances, understanding molecular fractions helps us study the composition and properties of complex molecular structures. These analyses are essential in fields ranging from chemistry to pharmacology.