The mole

1. Overview

The amount of substance is a fundamental physical property that quantifies the number of particles in a sample. In Physics, the mole acts as the essential mathematical bridge between the macroscopic properties of matter (such as the mass of a gas in a container) and the microscopic properties (such as the number of atoms or molecules). This quantification is vital for the study of thermodynamics, as the behavior of an ideal gas—including its pressure, volume, and temperature—is directly dependent on the number of constituent particles rather than the total mass of the substance.

Key Definitions

• Amount of substance ($n$): An SI base quantity that represents the number of elementary entities present in a sample. It is distinct from mass, as it counts the number of particles rather than measuring the quantity of matter or inertia.
• Mole (mol): The SI base unit for the amount of substance. One mole is defined as the amount of substance that contains exactly $6.02 \times 10^{23}$ elementary entities.
• Avogadro constant ($N_A$): The number of elementary entities per mole of substance. Its value is approximately $6.02 \times 10^{23} \text{ mol}^{-1}$. It serves as the proportionality constant between the number of particles and the amount of substance.
• Elementary Entity: The specific type of particle being counted. This must be specified depending on the context and can be an atom, a molecule, an ion, or an electron.
• Molar Mass ($M$): The mass of one mole of a substance. In the SI system, its unit is $\text{kg mol}^{-1}$, though it is frequently encountered in $\text{g mol}^{-1}$ in practical problems.
• Unified Atomic Mass Unit ($u$): A small unit of mass used to express atomic and molecular weights, defined as $1/12$ of the mass of an atom of carbon-12. $1 \text{ u} \approx 1.66 \times 10^{-27} \text{ kg}$.

Content

The Mole as an SI Base Quantity

The International System of Units (SI) identifies seven base quantities from which all other units are derived. The amount of substance is one of these seven, and the mole is its corresponding base unit.

In A-Level Physics, it is critical to understand that "amount of substance" is not a synonym for "mass."

1. Mass ($m$) is measured in kilograms (kg) and relates to the total quantity of matter.
2. Amount of substance ($n$) is measured in moles (mol) and relates to the total number of particles.

For example, 1 mole of Hydrogen gas ($H_2$) and 1 mole of Helium gas ($He$) contain the exact same number of particles ($6.02 \times 10^{23}$), but the Hydrogen sample has a mass of approximately $2 \text{ g}$ while the Helium sample has a mass of approximately $4 \text{ g}$.

The Avogadro Constant ($N_A$)

The Avogadro constant is the link between the macroscopic world of the laboratory and the microscopic world of atoms.

$N_A = 6.02 \times 10^{23} \text{ mol}^{-1}$

This constant defines the number of particles in one mole. Whether you are dealing with a mole of electrons, a mole of photons, or a mole of lead atoms, the number of entities remains constant. In the Cambridge 9702 syllabus, you should always use the value provided on the Data Sheet ($6.02 \times 10^{23}$) to ensure your calculations align with the mark scheme.

The Concept of the "Elementary Entity"

A common source of error in exams is failing to identify the "elementary entity." The mole counts whatever unit you define.

• In a monatomic gas like Argon ($Ar$), the elementary entity is the atom.
• In a diatomic gas like Nitrogen ($N_2$), the elementary entity is the molecule.
• If a question asks for the number of atoms in a mole of Nitrogen gas, you must account for the fact that each molecule contains two atoms. Therefore, 1 mole of $N_2$ molecules contains $2 \times N_A$ atoms.

Relating Particles, Moles, and Mass

To solve problems in Thermal Physics, you must be able to convert between the number of particles ($N$), the amount of substance ($n$), and the mass of the sample ($m$).

1. Converting between Moles and Number of Particles: The total number of particles $N$ is the product of the number of moles and the number of particles per mole.

$N = n N_A$

2. Converting between Moles and Mass: The amount of substance is the ratio of the total mass of the sample to the mass of a single mole.

$n = \frac{m}{M}$

3. The Mass of a Single Particle ($m_p$): Physicists often need the mass of one individual atom or molecule. This is found by dividing the molar mass by the Avogadro constant.

$m_p = \frac{M}{N_A}$

Note: If $M$ is in $\text{kg mol}^{-1}$, $m_p$ will be in $\text{kg}$.

Molar Mass and the Periodic Table

The molar mass $M$ of an element is numerically equal to its relative atomic mass ($A_r$) or relative molecular mass ($M_r$), but it is expressed in units of $\text{g mol}^{-1}$.

• Example: Carbon has an $A_r$ of 12.0. Its molar mass $M$ is $12.0 \text{ g mol}^{-1}$.
• In SI units (required for most physics equations like $pV = nRT$), this must be converted to $\text{kg mol}^{-1}$.
• $12.0 \text{ g mol}^{-1} = 0.012 \text{ kg mol}^{-1}$.

The Microscopic-Macroscopic Bridge in Ideal Gases

The mole is the foundation for the two forms of the Ideal Gas Equation:

1. $pV = nRT$ (using moles $n$ and the Molar Gas Constant $R$)
2. $pV = NkT$ (using number of particles $N$ and the Boltzmann Constant $k$)

The relationship between these two constants is defined by the Avogadro constant: $R = N_A k$

This demonstrates that the mole is not just a convenience for chemists, but a fundamental scaling factor in the laws of physics.

---

Worked Example 1 — Calculating Number of Atoms in a Diatomic Gas

A container holds $14.0 \text{ g}$ of Nitrogen gas ($N_2$). The molar mass of Nitrogen atoms is $14.0 \text{ g mol}^{-1}$. Calculate the total number of atoms present in the container.

Step 1: Determine the molar mass of the molecule ($N_2$). Since Nitrogen is diatomic, $M = 2 \times 14.0 = 28.0 \text{ g mol}^{-1}$.

Step 2: Calculate the amount of substance ($n$) in moles. $$n = \frac{m}{M}$$ $$n = \frac{14.0 \text{ g}}{28.0 \text{ g mol}^{-1}} = 0.500 \text{ mol}$$

Step 3: Calculate the number of molecules ($N_{molecules}$). $$N_{molecules} = n \times N_A$$ $$N_{molecules} = 0.500 \times (6.02 \times 10^{23}) = 3.01 \times 10^{23} \text{ molecules}$$

Step 4: Calculate the number of atoms. Each molecule of $N_2$ contains 2 atoms. $$N_{atoms} = 2 \times N_{molecules}$$ $$N_{atoms} = 2 \times 3.01 \times 10^{23} = 6.02 \times 10^{23} \text{ atoms}$$

Answer: $6.02 \times 10^{23}$ atoms

---

Worked Example 2 — Finding the Mass of a Single Atom

The molar mass of Gold ($Au$) is $197 \text{ g mol}^{-1}$. Calculate the mass of a single gold atom in kilograms.

Step 1: Convert molar mass to SI units ($\text{kg mol}^{-1}$). $$M = 197 \text{ g mol}^{-1} = 0.197 \text{ kg mol}^{-1}$$

Step 2: Use the relationship between $M$, $N_A$, and the mass of one particle ($m_p$). $$m_p = \frac{M}{N_A}$$ $$m_p = \frac{0.197}{6.02 \times 10^{23}}$$

Step 3: Calculate the final value. $$m_p = 3.2724 \times 10^{-25} \text{ kg}$$

Answer: $3.27 \times 10^{-25} \text{ kg}$ (to 3 s.f.)

---

Worked Example 3 — Moles and Gas Density

A sample of an unknown monatomic gas has a density of $1.60 \text{ kg m}^{-3}$ at a volume of $0.500 \text{ m}^3$. If the sample contains $20.0 \text{ moles}$ of the gas, identify the gas by calculating its molar mass.

Step 1: Calculate the total mass ($m$) of the gas. $$\text{mass} = \text{density} \times \text{volume}$$ $$m = 1.60 \times 0.500 = 0.800 \text{ kg}$$

Step 2: Calculate the molar mass ($M$) using the amount of substance ($n$). $$n = \frac{m}{M} \implies M = \frac{m}{n}$$ $$M = \frac{0.800 \text{ kg}}{20.0 \text{ mol}} = 0.040 \text{ kg mol}^{-1}$$

Step 3: Convert to $\text{g mol}^{-1}$ to compare with the periodic table. $$M = 40.0 \text{ g mol}^{-1}$$

Answer: The molar mass is $40.0 \text{ g mol}^{-1}$. (The gas is Argon).

---

Key Equations

Equation	Symbols	Status
$N = n N_A$	$N$: Number of particles $n$: Amount of substance (mol) $N_A$: Avogadro constant	Memorise
$n = \frac{m}{M}$	$n$: Amount of substance (mol) $m$: Mass of sample (kg or g) $M$: Molar mass (kg/mol or g/mol)	Memorise
$m_p = \frac{M}{N_A}$	$m_p$: Mass of one particle (kg) $M$: Molar mass (kg/mol) $N_A$: Avogadro constant	Derive/Memorise
$N_A = 6.02 \times 10^{23} \text{ mol}^{-1}$	$N_A$: Avogadro constant	Data Sheet

---

Common Mistakes to Avoid

• ❌ Wrong: Confusing the symbols $n$ and $N$.
  • ✓ Right: $n$ is the amount of substance (a small number, e.g., 2.5 mol). $N$ is the number of particles (a huge number, e.g., $1.5 \times 10^{24}$).
• ❌ Wrong: Using grams in the Ideal Gas Law ($pV=nRT$).
  • ✓ Right: While $n = m/M$ works if both are in grams, the pressure and volume units in $pV=nRT$ usually require SI units. It is safest to convert $M$ to $\text{kg mol}^{-1}$ immediately.
• ❌ Wrong: Forgetting the factor of 2 for diatomic gases.
  • ✓ Right: If the question mentions "Oxygen gas" or "Nitrogen gas," the particles are $O_2$ or $N_2$. If you need the number of atoms, you must multiply the number of molecules by 2.
• ❌ Wrong: Thinking the mole is a measure of mass.
  • ✓ Right: The mole is a measure of count. One mole of lead has much more mass than one mole of helium, even though they have the same number of atoms.
• ❌ Wrong: Using $N$ in the equation $pV = nRT$.
  • ✓ Right: If you use the number of particles $N$, you must use the Boltzmann constant $k$ ($pV = NkT$). If you use moles $n$, you must use the molar gas constant $R$ ($pV = nRT$).

---

Exam Tips

1. Check the Unit of $M$: In Physics papers, molar mass is often given in $\text{g mol}^{-1}$. Always check if you need to convert this to $10^{-3} \text{ kg mol}^{-1}$ before plugging it into equations involving Joules, Pascals, or Newtons.
2. Significant Figures: The Avogadro constant on the data sheet is $6.02 \times 10^{23}$ (3 s.f.). Ensure your final answers are rounded to 2 or 3 significant figures to match the precision of the provided constants.
3. The "Mole-Particle" Logic: If a question asks for the "number of particles," your answer should almost always have a large positive power of 10 (usually $10^{22}$ to $10^{26}$). If you get a small number, you've likely calculated the number of moles ($n$) instead.
4. Base Quantity Questions: If an exam question asks you to state the SI base unit for amount of substance, the answer is mol. If it asks for the SI base quantity, the answer is amount of substance.
5. Read the "Entity" Carefully: Does the question ask for the number of molecules, the number of atoms, or the number of electrons? In a gas like $CO_2$, one mole contains $N_A$ molecules, but $3 \times N_A$ atoms.
6. Data Sheet Reliance: Always use the value of $N_A$ from the data sheet provided in the exam booklet, even if you have memorized a more precise version (like $6.022 \times 10^{23}$). This ensures your intermediate steps match the mark scheme's range.