56 Molar Mass

Avogadro’s Number and the Mole

The mole is represented by Avogadro’s number, which is 6.022×10²³ atoms or molecules per mol.

The chemical changes observed in any reaction involve the rearrangement of billions of atoms. It is impractical to try to count or visualize all these atoms, but scientists need some way to refer to the entire quantity. They also need a way to compare these numbers and relate them to the weights of the substances, which they can measure and observe. The solution is the concept of the mole, which is very important in quantitative chemistry.

Avogadro’s Number

Amadeo Avogadro first proposed that the volume of a gas at a given pressure and temperature is proportional to the number of atoms or molecules,

regardless of the type of gas. Although he did not determine the exact proportion, he is credited for the idea.

Avogadro’s number is a proportion that relates molar mass on an atomic scale to physical mass on a human scale. Avogadro’s number is defined as the number of elementary particles (molecules, atoms, compounds, etc.) per mole of a substance. It is equal to 6.022×10²³ mol^-1 and is expressed as the symbol N_A.

Avogadro’s number is a similar concept to that of a dozen or a gross. A dozen molecules is 12 molecules. A gross of molecules is 144 molecules. Avogadro’s number is 6.022×10²³ molecules. With Avogadro’s number, scientists can discuss and compare very large numbers, which is useful because substances in everyday quantities contain very large numbers of atoms and molecules.

The Mole

The mole (abbreviated mol) is the SI measure of quantity of a “chemical entity,” such as atoms, electrons, or protons. It is defined as the amount of a substance that contains as many particles as there are atoms in 12 grams of pure carbon-12. So, 1 mol contains 6.022×10²³ elementary entities of the substance.

Chemical Computations with Avogadro’s Number and the Mole

Avogadro’s number is fundamental to understanding both the makeup of molecules and their interactions and combinations. For example, since one atom of oxygen will combine with two atoms of hydrogen to create one molecule of water ([latex]\text{H}_2\text{O}[/latex]), one mole of oxygen (6.022×10²³ of O atoms) will combine with two moles of hydrogen (2 × 6.022×10²³ of H atoms) to make one mole of [latex]\text{H}_2\text{O}[/latex].

Another property of Avogadro’s number is that the mass of one mole of a substance is equal to that substance’s molecular weight. For example, the mean molecular weight of water is 18.015 atomic mass units (amu), so one mole of water weight 18.015 grams. This property simplifies many chemical computations.

If you have 1.25 grams of a molecule with molecular weight of 134.1 g/mol, how many moles of that molecule do you have?

[latex]\text{1.25 g} \times \frac{\text{1 mole}}{\text{134.1 g}} = \text{0.0093 moles}[/latex]

The Mole, Avogadro: This video introduces counting by mass, the mole, and how it relates to atomic mass units (AMU) and Avogadro’s number.

Converting between Moles and Atoms

By understanding the relationship between moles and Avogadro’s number, scientists can convert between number of moles and number of atoms.

Moles and Atoms

As introduced in the previous concept, the mole can be used to relate masses of substances to the quantity of atoms therein. This is an easy way of determining how much of one substance can react with a given amount of another substance.

From moles of a substance, one can also find the number of atoms in a sample and vice versa. The bridge between atoms and moles is Avogadro’s number, 6.022×10²³.

Avogadro’s number is typically dimensionless, but when it defines the mole, it can be expressed as 6.022×10²³ elementary entities/mol. This form shows the role of Avogadro’s number as a conversion factor between the number of entities and the number of moles. Therefore, given the relationship 1 mol = 6.022 x 10²³ atoms, converting between moles and atoms of a substance becomes a simple dimensional analysis problem.

Converting Moles to Atoms

Given a known number of moles (x), one can find the number of atoms (y) in this molar quantity by multiplying it by Avogadro’s number:

[latex]x \text{ moles} \cdot \frac{6.022 \times 10^{23} \text{atoms}}{\text{1 mole}} = y \text{ atoms}[/latex]

For example, if scientists want to know how may atoms are in six moles of sodium (x = 6), they could solve:

[latex]\text{6 moles} \cdot \frac {6.022 \times 10^{23} \text{atoms} }{\text{1 mole}} = 3.61 \times 10^{24} \text{ atoms}[/latex]

Note that the solution is independent of whether the element is sodium or otherwise.

Converting Atoms to Moles

Reversing the calculation above, it is possible to convert a number of atoms to a molar quantity by dividing it by Avogadro’s number:

[latex]\frac{ x \text{ atoms} }{ 6.022 \times 10^{23} \frac{ \text{ atoms} }{ \text{1 mole} } } = y \text{ moles}[/latex]

This can be written without a fraction in the denominator by multiplying the number of atoms by the reciprocal of Avogadro’s number:

[latex]x \text{ atoms} \cdot \frac{\text{1 mole}}{6.022 \times 10^{23}} = y \text{ moles}[/latex]

For example, if scientists know there are [latex]3.5 \cdot 10^{24}[/latex] in a sample, they can calculate the number of moles this quantity represents:

[latex]3.5 \times 10^{24} \text{atoms} \cdot \frac{\text{1 mole}}{6.022 \times 10^{23} \text{atoms} } = 5.81 \text{ moles}[/latex]

Molar Mass of Compounds

The molar mass of a particular substance is the mass of one mole of that substance.

Measuring Mass in Chemistry

Chemists can measure a quantity of matter using mass, but in chemical reactions it is often important to consider the number of atoms of each element present in each sample. Even the smallest quantity of a substance will contain billions of atoms, so chemists generally use the mole as the unit for the amount of substance.

One mole (abbreviated mol) is equal to the number of atoms in 12 grams of carbon-12; this number is referred to as Avogadro’s number and has been measured as approximately 6.022 x 10²³. In other words, a mole is the amount of substance that contains as many entities (atoms, or other particles) as there are atoms in 12 grams of pure carbon-12.

amu vs. g/mol

Each ion, or atom, has a particular mass; similarly, each mole of a given pure substance also has a definite mass. The mass of one mole of atoms of a pure element in grams is equivalent to the atomic mass of that element in atomic mass units (amu) or in grams per mole (g/mol). Although mass can be expressed as both amu and g/mol, g/mol is the most useful system of units for laboratory chemistry.

Calculating Molar Mass

Molar mass is the mass of a given substance divided by the amount of that substance, measured in g/mol. For example, the atomic mass of titanium is 47.88 amu or 47.88 g/mol. In 47.88 grams of titanium, there is one mole, or 6.022 x 10²³ titanium atoms.

The characteristic molar mass of an element is simply the atomic mass in g/mol. However, molar mass can also be calculated by multiplying the atomic mass in amu by the molar mass constant (1 g/mol). To calculate the molar mass of a compound with multiple atoms, sum all the atomic mass of the constituent atoms.

For example, the molar mass of [latex]\text{NaCl}[/latex] can be calculated for finding the atomic mass of sodium (22.99 g/mol) and the atomic mass of chlorine (35.45 g/mol) and combining them. The molar mass of [latex]\text{NaCl}[/latex] is 58.44 g/mol.

Molar Mass Calculations – YouTube: This video shows how to calculate the molar mass for several compounds using their chemical formulas.

Converting between Mass and Number of Moles

A substance’s molar mass can be used to convert between the mass of the substance and the number of moles in that substance.

Chemists generally use the mole as the unit for the number of atoms or molecules of a material. One mole (abbreviated mol) is equal to 6.022×10²³ molecular entities (Avogadro’s number), and each element has a different molar mass depending on the weight of 6.022×10²³ of its atoms (1 mole). The molar mass of any element can be determined by finding the atomic mass of the element on the periodic table. For example, if the atomic mass of sulfer (S) is 32.066 amu, then its molar mass is 32.066 g/mol.

By recognizing the relationship between the molar mass (g/mol), moles (mol), and particles, scientists can use dimensional analysis convert between mass, number of moles and number of atoms very easily.

Determining the Molar Mass of a Compound

In a compound of [latex]\text{NaOH}[/latex], the molar mass of Na alone is 23 g/mol, the molar mass of O is 16 g/mol, and H is 1 g/mol. What is the molar mass of [latex]\text{NaOH}[/latex]?

[latex]\text{Na} + \text{O} + \text{H} = \text{NaOH}[/latex]

[latex]\text{23 g/mol} + \text{16 g/mol} + \text{1 g/mol} = \text{40 g/mol}[/latex]

The molar mass of the compound [latex]\text{NaOH}[/latex] is 40 g/mol.

Converting Mass to Number of Moles

How many moles of [latex]\text{NaOH}[/latex] are present in 90 g of [latex]\text{NaOH}[/latex]?

Since the molar mass of [latex]\text{NaOH}[/latex] is 40 g/mol, we can divide the 90 g of [latex]\text{NaOH}[/latex] by the molar mass (40 g/mol) to find the moles of [latex]\text{NaOH}[/latex]. This the same as multiplying by the reciprocal of 40 g/mol.

If the equation is arranged correctly, the mass units (g) cancel out and leave moles as the unit.

[latex]\text{90 g NaOH} \times \frac{\text{1 mol}}{\text{40 g}} = \text{2.25 mol NaOH}[/latex]

There are 2.25 moles of [latex]\text{NaOH}[/latex] in 90g of [latex]\text{NaOH}[/latex].

Converting Between Mass, Number of Moles, and Number of Atoms

How many moles and how many atoms are contained in 10.0 g of nickel?

According to the periodic table, the atomic mass of nickel (Ni) is 58.69 amu, which means that the molar mass of nickel is 58.69 g/mol. Therefore, we can divide 10.0 g of Ni by the molar mass of Ni to find the number of moles present.

Using dimensional analysis, it is possible to determine that:

[latex]\text{10 g Ni} \times \frac{\text{1 mol Ni}}{\text{58.69 g Ni}} = \text{0.170 mol Ni}[/latex]

To determine the number of atoms, convert the moles of Ni to atoms using Avogadro’s number:

[latex]\text{0.170 moles Ni} \times \frac{6.022 \times 10^{23} \text{atoms Ni}}{\text{1 mol Ni}} = 1.02 \times 10^{23} \text{atoms Ni}[/latex]

Given a sample’s mass and number of moles in that sample, it is also possible to calculate the sample’s molecular mass by dividing the mass by the number of moles to calculate g/mol.

What is the molar mass of methane ([latex]\text{CH}_4[/latex]) if there are 0.623 moles in a 10.0g sample?

[latex]\frac{\text{10.0 g CH}_4}{\text{0.623 mol CH}_4} = \text{16.05 g/mol CH}_4[/latex]

The molar mass of [latex]\text{CH}_4[/latex] is 16.05 g/mol.