Problem 71 - General Chemistry: Principles and Modern Applications

Question

Use fundamental definitions and statements from Chapters 1 and 2 to establish the fact that \(6.022 \times 10^{23} \mathrm{u}=1.000 \mathrm{g}\)

Step-by-Step Solution

Answer

Knowing that 1 mole of any substance contains 6.022 x 10^23 entities (Avogadro's number) and the weight of 1 mole of that substance in grams is numerically equal to the atomic or molecular weight in u, we can establish that 6.022 x 10^23 u (Avogadro's number of atomic mass units) equals 1 gram.

Step 1: Define atomic mass unit (AMU)

One atomic mass unit (u) is defined as 1/12th the mass of a carbon-12 atom, which is approximately 1.66 x 10^-24 g.

Step 2: Apply the concept of a mole

One mole of any element or compound contains 6.022 x 10^23 entities (usually atoms or molecules), which is also known as Avogadro's number.

Step 3: Link AMU, grams and mole

When an element's atomic mass is expressed in u, it is almost the same number as the mass of one mole of that element in grams. For example, one mole of carbon-12 atoms has a mass of exactly 12 grams, because the atomic mass of carbon-12 is 12 u. Therefore, 1 u, as defined above, is equivalent to 1 g/mol.

Step 4: Establish the relationship

Applying these principles, we can say that 6.022 x 10^23 u (Avogadro's number of atomic mass units) is equivalent to 1 gram, because 1 mole of 1-u atoms weighs 1 gram.

Key Concepts

Atomic Mass Unit (AMU)

The atomic mass unit, or AMU, is a standard unit of mass that quantifies the amount of mass at an atomic or molecular scale. By definition, it is one twelfth of the mass of a carbon-12 atom. This specific choice comes because of carbon-12's natural abundance and stable structure. The AMU is approximately equal to 1.66 x 10^-24 grams.

Since atoms and molecules are incredibly small, dealing with their masses in grams would involve extremely tiny numbers, making calculations cumbersome. The atomic mass unit provides a convenient scale for expressing atomic and molecular masses. Whenever you see an element listed with a mass number on the periodic table, it is typically in atomic mass units.

Avogadro's Number

Avogadro's number is a fundamental constant of nature that is used to count large numbers of very small entities. It is exactly 6.022 x 10^{23}, and it's named after the scientist Amedeo Avogadro. This number represents the quantity of atoms or molecules in one mole of a substance.

The enormity of Avogadro's number makes it suitable for connecting the micro world of atoms to the macro world we experience. When you work with moles, you're making use of Avogadro's number to bridge the gap between immensely small particles and quantities you can handle in a lab setting.

Molar Mass

Molar mass is the mass of one mole of a given substance. It is usually measured in grams per mole (g/mol). To find the molar mass of an element, you simply take its atomic mass from the periodic table, which is expressed in atomic mass units, and state it in grams. This direct conversion is possible because of the definitions of both the atomic mass unit and Avogadro's number.

For example, one mole of carbon-12, with an atomic mass of 12 AMU, will weigh exactly 12 grams. Understanding molar mass is essential for converting between units of atoms, molecules, and grams in chemical calculations. It is a bridge between the atomic world and the tangible world of grams and moles.

Carbon-12 Atom

The carbon-12 atom plays a critical role in chemistry and physics because it serves as the standard for defining the atomic mass unit. Carbon-12 is an isotope of carbon, consisting of 6 protons and 6 neutrons, making it a stable and abundant form of carbon.

By assigning the carbon-12 atom a precise mass of 12 atomic mass units, scientists can effectively create a benchmark by which the masses of other atoms are measured. This standardization simplifies scientific communication and computation when dealing with atomic masses and molecular weights across the periodic table.
Understanding carbon-12's role enhances comprehension of the atomic mass unit and the concept of molar mass.