Avogadro’s Number Formula: Definition, Derivation, Examples

Avogadro’s Number Formula: Avogadro’s number, sometimes referred to as Avogadro’s constant, is a fundamental concept in the field of chemistry. It honors the contributions of Amedeo Avogadro, an influential scientist in the field. This number represents the incredible scale of atoms, molecules, or ions in just one mole of any substance, and it’s approximately equal to 6.022 x 10^23.

To put it in simpler terms, imagine you have a big bag of marbles, and each marble represents an atom or a molecule. Avogadro’s number is like telling you that in one mole of marbles (which is a standard amount in chemistry), you would have this enormous number of marbles, about 6.022 x 10^23 of them! This concept helps chemists work with chemicals more precisely and understand how they react with each other.

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Defining Avogadro’s Number

Avogadro’s number, also known as Avogadro’s constant, denotes the number of atoms, molecules, and ions found in one gram of an element’s atoms, one gram of a compound’s molecules, or one gram of a substance’s ions, respectively. Another perspective defines it as the quantity of atoms present in 12 grams of the C-12 isotope. Avogadro’s number is represented by Na, where its numerical value is 6.022 × 10^23 particles per mole.

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The Significance of Avogadro’s Constant

In chemistry, substances are commonly assessed using atomic mass units (amu), defined as one-twelfth of the mass of a carbon atom. For instance, hydrogen’s atomic mass unit is 1.00794 amu. However, determining a particle’s reactivity, such as an atom or molecule, in terms of atomic mass units is challenging. To address this, chemists have established a relationship between atomic mass units and grams, setting 1 amu equivalent to 1.66 x 10^-24 grams. This conversion factor facilitates the seamless conversion between gram measurements and the abstract unit of measurement used by atomic mass units, underscoring the importance of Avogadro’s number.

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The Determination of Avogadro’s Number

Accurately determining Avogadro’s number necessitates measuring a single quantity on both atomic and macroscopic scales using a consistent unit of measurement. This achievement became possible through the work of American physicist Robert Millikan, who measured the charge on an electron. The charge on a mole of electrons had previously been established as the Faraday constant, with a best estimate of 96,485.3383 coulombs per mole of electrons according to the National Institute of Standards and Technology (NIST). Modern experiments have refined this value, pinpointing the charge on a single electron at 1.60217653 x 10^-19 coulombs per electron. Dividing the Faraday constant by the charge on a single electron yields a precise value for Avogadro’s number, which stands at 6.02214154 x 10^23 particles per mole.

The Formula for Avogadro’s Number

The formula to calculate Avogadro’s Number is straightforward:

Avogadro’s Number = 6.022 × 10^23 Particles

One Mole of Substance = 6.022 × 10^23 Particles

Several Examples illustrate Avogadro’s Number

1.008 grams of hydrogen atoms = 1 mole of hydrogen = 6.022 × 10^23 atoms of hydrogen.

23 grams of sodium atoms = 1 mole of sodium = 6.022 × 10^23 atoms of sodium.

18 grams of water = 1 mole of water = 6.022 × 10^23 molecules of water.

In essence, 6.022 × 10^23 particles equate to one mole of the substance.

Derivation of Avogadro’s Number Formula

An atomic mass unit (a.m.u) is equivalent to 1.66 × 10^-24 grams.

One mole of a substance is defined as having the same number of entities (like atoms) as there are atoms in exactly 12 grams of the carbon-12 isotope.

The mass of a single carbon-12 atom is calculated as 12 multiplied by 1.66 × 10^-24 grams, resulting in 1.992 × 10^-23 grams.

Consequently, since one mole of carbon-12 atoms weighs 12 grams, it contains a staggering 6.0221367 × 10^23 atoms per mole.

Utilization of Avogadro’s Number

Avogadro’s number holds various practical applications:

Relation to Gay-Lussac’s Law: Avogadro’s number is closely intertwined with Gay-Lussac’s law, resolving issues in the law by suggesting that equal volumes of gases at the same temperature and pressure contain an equal number of molecules.

Mass-Volume Relationship in Gases: It plays a pivotal role in establishing the relationship between the mass and volume of gases. According to this concept, at constant temperature and pressure, equal volumes of gases contain the same number of molecules, regardless of the gas’s identity.

Determining Atomicity: Avogadro’s number aids in determining the atomicity of gases, revealing the number of atoms within a molecule of a substance. For instance, oxygen (O2) has an atomicity of 2 as it consists of two oxygen atoms, while sulfur (S8) has an atomicity of 8, containing eight sulfur atoms.

Vapour Density: Avogadro’s number assists in establishing a relationship between the molecular mass and the vapor density of a gas. Vapor density is defined as the ratio of a gas’s mass to its volume at specific temperature and pressure. This relationship is expressed as Vapor Density (V.D) = Relative molecular mass / 2 or Relative molecular mass = 2 x Vapor Density.

The Significance of Avogadro’s Number

Avogadro’s number bridges the gap between the macroscopic and microscopic worlds, providing a link between various physical constants and their characteristics. It connects:

Gas constant (R) and Boltzmann constant (kA) through R = NAkA.

Faraday constant (F) and electron charge (e) through F = eNA.

Atomic Mass Constant (Mu) and Molar Mass Constant (1u) through Mu = NA(1u).

Solved Examples of Avogadro’s Number

Example 1: Calculate the mass of (i) an atom of silver (ii) a molecule of carbon dioxide.

Solution:

(i) Atomic mass of silver = 108 u

1 mole of Ag atom = 108 g = 6.022×10^23 atoms

Mass of one atom of silver = 108 / 6.022×10^23 = 1.793×10^-22 g

(ii) Molecular mass of CO2 = 1×12 + 2×16 = 44 u

1 mole of CO2 = 44 g = 6.022×10^23 atoms

Mass of one molecule of CO2 = 44 / 6.022×10^23 = 7.307×10^-23 g

Example 2: Determine the number of atoms and molecules of sulfur in 64.0 g of sulfur (S8).

Solution:

Molecular mass of S8 = 32 × 8 = 256 u

1 mole of sulfur molecules = 256 g = 6.022 × 10^23 molecules of sulfur

1 gram of sulfur molecules = 6.022 × 10^23 / 256 = 0.023523 × 10^23 molecules

64 grams of sulfur molecules = 64 × 0.023523 × 10^23 = 1.505 × 10^23 molecules

1 molecule of sulfur S8 contains 8 sulfur atoms

1.505 × 10^23 molecules