Revise The mole for Cambridge 9702 (15.1)

The 'amount of substance' is a base quantity in physics, and its SI unit is the mole, symbol 'mol' (or sometimes 'n' in equations). One mole is precisely defined as the quantity of substance that contains the same number of elementary entities (like atoms, molecules, or ions) as there are atoms in exactly 12 grams of the Carbon-12 isotope.

This specific number of particles within one mole of any substance is known as the Avogadro constant, symbol $N_A$. Its accepted value is approximately $6.02 \times 10^{23} \text{ mol}^{-1}$. This means whether you have one mole of hydrogen atoms or one mole of water molecules, you'll always have $6.02 \times 10^{23}$ of those respective particles.

$N = n N_A$

We can derive the value of the Avogadro constant from the definition of the mole. <br>1. The mass of one Carbon-12 atom is defined as 12 atomic mass units (u). <br>2. One atomic mass unit (u) is approximately $1.6605 \times 10^{-27} \text{ kg}$. <br>3. Therefore, the mass of one Carbon-12 atom is $12 \times 1.6605 \times 10^{-27} \text{ kg} = 1.9926 \times 10^{-26} \text{ kg}$. <br>4. The number of atoms in 12g (0.012 kg) of Carbon-12 is the total mass divided by the mass of one atom: $N_A = \frac{0.012 \text{ kg}}{1.9926 \times 10^{-26} \text{ kg}} \approx 6.022 \times 10^{23}$. <br>This confirms the value of the Avogadro constant.

The molar mass (M) of a substance is the mass of one mole of that substance. It provides a direct link between the macroscopic mass of a sample (what you can weigh on a scale) and the number of moles it contains. The SI unit for molar mass is kg/mol, but it is often expressed in g/mol in chemistry. For example, the molar mass of Carbon-12 is exactly 12 g/mol by definition.

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

The concept of moles is vital when studying gases, particularly with the Ideal Gas Equation. This equation describes the behaviour of an ideal gas and uses 'n' to represent the number of moles of the gas. Remember that temperature ($T$) must always be in Kelvin for gas law calculations ($T_K = T_C + 273$).

$PV = nRT$

The Ideal Gas Equation can also be expressed using the total number of individual molecules, $N$, instead of moles. This form involves the Boltzmann constant, $k$, which links energy to temperature at the molecular level. It's essentially the gas constant per individual particle.

$PV = NkT$

$k = R / N_A$