Question

Suppose that the atomic mass of \(\mathrm{C}-12\) is taken to be \(5.000 \mathrm{amu}\) and that a mole is defined as the number of atoms in \(5.000 \mathrm{~kg}\) of carbon-12. How many atoms would there be in one mole under these conditions? (Hint: There are \(6.022 \times 10^{23} \mathrm{C}\) atoms in \(12.00 \mathrm{~g}\) of \(\mathrm{C}-12 .\) )

Short answer

Answer: Under these conditions, there would be 6.022 x 10^26 atoms in one modified mole.

Step-by-step solution

Convert atomic mass and mass of carbon-12 to grams

We will first convert given atomic mass (5 amu) of C-12 to grams (g) and convert the mass of carbon-12 for the modified definition of mole (5 kg) to grams: Atomic mass of C-12 = \(5 \mathrm{~amu} \times \frac{1 \mathrm{~g}}{1\mathrm{~amu}} = 5\mathrm{~g}\) (We ignore the actual atomic mass conversion since it is given that C-12 has an atomic mass of 5 amu) Mass of carbon-12 for modified mole = \(5.000 \mathrm{~kg} \times \frac{1000 \mathrm{~g}}{1 \mathrm{~kg}} = 5000 \mathrm{~g}\)

Find the ratio between modified mole and typical mole

Using the hint, we know that there are \(6.022 \times 10^{23}\) C atoms in \(12.00 \mathrm{~g}\) of C-12. Since the atomic mass of C-12 under these conditions is 5g, there will be that many atoms in \(5\mathrm{~g}\) of C-12 under these conditions as well. Next, we find the ratio between 5000g of C-12 (modified mole definition) and 5g of C-12: Ratio = \(\frac{5000\mathrm{~g}}{5\mathrm{~g}} = 1000\)

Find the number of atoms in one modified mole

To find the number of atoms in one mole under these conditions, we multiply the typical number of atoms in a mole (\(6.022 \times 10^{23}\)) by the ratio we found earlier (1000): Number of atoms in one modified mole = \(6.022 \times 10^{23} \times 1000 = 6.022 \times 10^{26}\) So, under these conditions, there would be \(6.022 \times 10^{26}\) atoms in one mole.

Key concepts

Atomic Mass Unit (amu)

Understanding atomic mass unit (amu) is pivotal in grasping the essence of chemical measurements. The amu is a standard unit of mass that quantifies the mass of atoms or molecules. It is defined as one twelfth of the mass of an unbound neutral atom of carbon-12 at rest and in its ground state. This precise measure, equivalent to approximately 1.66053906660 x 10^-24 grams, allows chemists to express atomic and molecular masses in a consistent manner.

When we refer to the atomic mass of carbon-12 as '5 amu' in the exercise, it's hypothetical, designed to guide students through a problem-solving process. In reality, the actual atomic mass of any element, including carbon-12, is a constant value based on the defined standard of 'amu'. Every element on the periodic table has a specific atomic mass measured in amu, which corresponds to the weighted average of the isotopes that exist for that element.

Avogadro's Number

Avogadro's number, symbolized as N_A, is a constant that represents the number of constituent particles, typically atoms or molecules, that are contained in one mole of a substance. This number is approximately 6.022 x 10²³ particles per mole and is essential for converting between the number of particles and the amount of substance measured in moles.

The concept of the mole is a bridge between the microscopic world of atoms and the macroscopic world we experience, and Avogadro's number serves as that connector. It enables chemists to count atoms by weighing out a mass of a substance that corresponds to its molar mass in grams. In the exercise, using Avogadro's number aids in determining how many atoms would be present in a modified mole of carbon-12 under unconventional definitions.

Carbon-12 Isotopic Mass

The isotopic mass of carbon-12 is notably significant because it serves as the reference from which the atomic mass unit is derived. An isotope is an atom of an element that has the same number of protons but a different number of neutrons, resulting in a different mass. Carbon-12 is the isotope of carbon with six protons and six neutrons in its nucleus.

The isotopic mass of carbon-12 is set as exactly 12 atomic mass units (12 amu), by convention, making it the calibration standard for atomic mass units and molar mass calculations. In this simplified exercise, the isotopic mass is altered to 5 amu to illustrate the relationship between atomic mass and mole calculations. It stresses the need to understand how molar mass links the microscopic properties of atoms to the macroscopic measurements chemists perform in the laboratory.