Problem 51 - Introductory Chemistry Atoms First

Question

How many atoms of the \({ }_{6}^{12} \mathrm{C}\) isotope are there in exactly \(12 \mathrm{~g}\) of the isotope?

Step-by-Step Solution

Answer

There are approximately \(6.022 \times 10^{23}\) atoms of the \({ }_{6}^{12} \mathrm{C}\) isotope in exactly 12g of the isotope.

Step 1: Determine the molar mass of the given isotope

To determine the molar mass of the \({ }_{6}^{12} \mathrm{C}\) isotopes, we can look at the superscript, which represents the mass number. In this case, it is 12. Therefore, the molar mass of the \({ }_{6}^{12} \mathrm{C}\) isotope is 12g/mol.

Step 2: Calculate the number of moles of the isotope

Given the mass of the isotope (12 g) and the molar mass of the isotope (12 g/mol), we can calculate the number of moles of the isotope as follows: Number of moles = (mass of isotope) / (molar mass of isotope) Number of moles = (12 g) / (12 g/mol) = 1 mole

Step 3: Determine the number of atoms using Avogadro's number

We know that one mole of any substance contains Avogadro's number of particles or atoms, which is approximately \(6.022 \times 10^{23}\) atoms/mol. So, to find the number of atoms in 1 mole of the given isotope, we can use this formula: Number of atoms = (number of moles) × (Avogadro's number) Number of atoms = (1 mole) × (6.022 × 10^23 atoms/mol) = 6.022 x 10^23 atoms

Step 4: Conclusion

There are approximately \(6.022 \times 10^{23}\) atoms of the \({ }_{6}^{12} \mathrm{C}\) isotope in exactly 12g of the isotope.

Key Concepts

Avogadro's Number

When measuring the amount of a substance in chemistry, we use a special counting unit known as the mole, which is connected to a fundamental constant called Avogadro's number. This is an incredibly large number, quantified as approximately \(6.022 \times 10^{23}\) entities per mole. Whether you’re counting atoms, molecules, or ions, this number remains constant for one mole of any substance.

Think of Avogadro's number as the chemistry equivalent of a dozen, but instead of 12 items in a dozen, there are \(6.022 \times 10^{23}\) entities in a mole. This number is derived from the number of atoms found in exactly 12 grams of carbon-12, which lays the foundation for understanding the mole concept. This relationship helps chemists relate mass to number of atoms, a pivotal step in many chemical calculations.

Molar Mass Calculation

The concept of molar mass is central to understanding the quantitative aspects of the mole concept. The molar mass is the weight of one mole of any chemical entities, including atoms, molecules, or ions, expressed in grams. It serves as a bridge between the atomic scale and the macro scale.

To calculate the molar mass, we simply add the atomic masses of all the atoms in a molecule, taking into account their respective quantities. For example, the atomic mass of carbon (C) is approximately 12 atomic mass units (amu). Thus, for the \( \_6^{12} \mathrm{C} \) isotope, the molar mass is simply 12 grams per mole - since this isotope consists of atoms each with a mass of 12 amu.

Atomic Mass Unit

An atomic mass unit (amu) is a small unit of mass used to express the mass of atoms and molecules. Precisely, 1 amu is defined as one twelfth of the mass of a carbon-12 atom. In terms of actual mass, 1 amu corresponds to \(1.66053906660 \times 10^{-24}\) grams.

By using amu, chemists can use a number that is much more manageable than dealing with grams when discussing and calculating the tiny masses of individual atoms or molecules. This helps in comparing the weights of different atoms and summarizing the weight of a single molecule from the combined amu's of its constituent atoms, as shown in the molar mass calculations.

Isotopes in Chemistry

Isotopes are variants of a particular chemical element that have the same number of protons, thus the same atomic number, but different numbers of neutrons, giving them different mass numbers. This results in isotopes having different physical properties despite having the same chemical properties.

For example, carbon-12 (\( \_6^{12} \mathrm{C} \) ) and carbon-13 (\( \_6^{13} \mathrm{C} \) ) are isotopes of carbon. They both have 6 protons, but carbon-12 has 6 neutrons whereas carbon-13 has 7 neutrons. This variance in mass among isotopes is key when calculating molar mass because each isotope of an element will contribute a slightly different weight, and understanding isotopes is essential when working with naturally occurring elements, which are usually a mixture of isotopes.