Avogadro, Molar Gas & Boltzmann Constant (AQA A Level Physics): Revision Note

Avogadro, molar gas & Boltzmann constant

Avogadro's constant

• The atomic mass unit (u) is approximately the mass of a proton or neutron = 1.66 × 10^-27 kg

• This means that an atom or molecule has a mass approximately equal to the number of protons and neutrons it contains

• A carbon-12 atom has a mass of:

$12\mathrm{u}=12\times 1.66\times {10}^{-27}=1.99\times {10}^{-26}\mathrm{kg}$

• The exact number for a mole is defined as the number of molecules in exactly 12 g of carbon:

$1\mathrm{mole}=\frac{0.012}{1.99\times {10}^{-26}}=6.02\times {10}^{23}\mathrm{molecules}$

• Avogadro’s constant ($N_A$) is defined as:

   The number of atoms of carbon-12 in 12 g of carbon-12; equal to 6.02 × 10²³ mol^-1

• For example, 1 mole of sodium (Na) contains 6.02 × 10²³ atoms of sodium

• The number of atoms (or molecules) can be determined if the number of moles is known by multiplying by $N_A$, for example:

2.0 mol of helium contains:  2.0 × N_A = 2.0 × 6.02 × 10²³ = 1.20 × 10²⁴ atoms

Moles and atomic mass

• One mole of any element is equal to the relative atomic mass of that element in grams

  • For example, helium has an atomic mass of 4, meaning 1 mole of helium has a mass of 4 g

• If the substance is a compound, add up the relative atomic masses, for example, water (H₂O) is made up of:

  • 2 hydrogen atoms (each with an atomic mass of 1) and 1 oxygen atom (atomic mass of 16)

  • So, 1 mole of water would have a mass of (2 × 1) + 16 = 18 g

Molar mass

• The molar mass of a substance is the mass, in grams, in one mole

  • Its unit is g mol^-1

• The number of moles from this can be calculated using the equation:

$\mathrm{Number}\mathrm{of}\mathrm{moles}\left(\mathrm{mol}\right)=\frac{\mathrm{mass}\left(\mathrm{g}\right)}{\mathrm{molar}\mathrm{mass}\left(\mathrm{g}{\mathrm{mol}}^{-1}\right)}$

Boltzmann and the molar gas constant

• The Boltzmann constant $k$ is used in the ideal gas equation and is defined by the equation:

$k=\frac{R}{N_A}$

• Where:

  • $R$ = molar gas constant

  • $N_A$ = Avogadro’s constant

• Boltzmann’s constant, therefore, has a value of:

$k=\frac{8.31}{6.02\times {10}^{23}}=1.38\times {10}^{-23}\mathrm{J}\hspace{0.17em}{\mathrm{K}}^{-1}$

• The Boltzmann constant relates the properties of microscopic particles (e.g. kinetic energy of gas molecules) to their macroscopic properties (e.g. temperature)

  • This is why the units are J K^-1

• Its value is very small because the increase in kinetic energy of a molecule is very small for every incremental increase in temperature