Chemistry · Fundamentals of chemistry · Grade 7

Mole, molar mass and volume

18 g of water and 2 g of hydrogen contain the same number of particles. Why? How much space does a mole of gas take up?

The situation

How many molecules are there in the water in a faceted glass in a teahouse — billions, trillions? Chemists do not count particles one by one: they count in moles — just as eggs are counted by the dozen and paper by the ream. At a party, helium balloons strive upwards, while a balloon filled with carbon dioxide sinks to the floor, although both have the same volume. And in the laboratory, by measuring the volume of hydrogen released from magnesium ribbon, one can find the equivalent of the metal. How much does one mole of a substance weigh, and how much space does one mole of a gas occupy?

Topic

Amount of substance: mole, molar mass, molar volume and equivalent. How many molecules are in a glass of water? From counting by the mole to a gas cylinder: 22.4 litres, a balloon of air and the volume of hydrogen released from magnesium ribbon

Key ideas

Avogadro constant
Molar mass
Molar volume

Formulas

n = \frac{m}{M} = \frac{N}{N_{A}}

Amount of substance (mol): $m$ is mass, $M$ is molar mass (g/mol), $N$ is the number of particles, $N_{A} = 6.022 \cdot 10^{23}$ mol⁻¹.

V = n V_{m}, V_{m} = 22.4 L/mol (STP)

From Avogadro's law: under standard conditions (0 °C, 101.325 kPa) 1 mol of any gas occupies about 22.4 L.

D_{2} (1) = \frac{M_{1}}{M_{2}}, D_{air} = \frac{M}{29}

Relative density: the ratio of the densities of two gases under the same conditions equals the ratio of their molar masses.

p V = n R T = \frac{m}{M} R T

The Mendeleev–Clapeyron equation: $R = 8.314$ J/(mol·K), $T$ in kelvins.

E = \frac{A}{valency}, E = \frac{m \cdot 11 200}{V_{0} (H_{2})}

Equivalent mass of a metal (g/mol): $V_{0}$ is the volume of hydrogen reduced to standard conditions, mL (1 mol-eq of H₂ = 11.2 L).

Theory

Amount of substance and the Avogadro constant

The amount of substance $n$ is a quantity that expresses the number of particles (atoms, molecules, ions) in a substance; its unit is the mole. 1 mol is exactly $N_{A} = 6.02214076 \cdot 10^{23}$ particles (since 2019 this number has been fixed as an exact value; before that 1 mol was defined as the number of atoms in 12 g of carbon-12). The mole is the chemist's “dozen”: just as we count eggs by the dozen, we count atoms in units of $6.02 \cdot 10^{23}$ . The formula: $n = N / N_{A}$ . A glass (250 mL) of water holds about 13.9 mol, that is $8.4 \cdot 10^{24}$ molecules.

Molar mass

Molar mass $M$ is the mass of 1 mol of a substance, in g/mol; its numerical value equals the relative molecular (atomic) mass: $M_{r} (H_{2} O) = 18.02$ → $M = 18.02$ g/mol. Hence $n = m / M$ : 18 g of water, 58.4 g of table salt, 342 g of sugar and 55.8 g of iron each contain the same number, $6.02 \cdot 10^{23}$ , of particles. Their masses and volumes are different: the particles themselves have different masses and are packed at different densities. The size of the “1 mol” samples in the scene is calculated from the real density.

Avogadro's law. Molar volume. Gas mixtures

Avogadro's law (1811): at the same temperature and pressure, equal volumes of any gases contain the same number of molecules. So 1 mol of any gas occupies the same volume — the molar volume: under standard conditions (STP: 0 °C, 101.325 kPa) $V_{m} \approx 22.4$ L/mol (more exactly 22.41); at 20 °C about 24.0 L/mol. The ratio of the densities of gases equals the ratio of their molar masses — the relative density $D$ . Air is a mixture (by volume N₂ 78.08 %, O₂ 20.95 %, Ar 0.93 %, CO₂ ≈ 0.04 %); its average molar mass is $\overline{M} = \sum φ_{i} M_{i} \approx 28.96 \approx 29$ g/mol. A gas with $M < 29$ (He, CH₄) is lighter than air — the balloon rises; with $M > 29$ (CO₂) it is heavier. This law does not apply to solids and liquids.

The Mendeleev–Clapeyron equation

Under any conditions, for an ideal gas $p V = n R T = \frac{m}{M} R T$ , $R = 8.314$ J/(mol·K). If pressure is taken in kPa and volume in litres, $R$ = 8.314 kPa·L/(mol·K). From it: the molar volume $V_{m} = R T / p$ (it increases with temperature and decreases with pressure), the mass of a gas $m = p V M / (R T)$ and the molar mass $M = m R T / (p V)$ . Laboratory (“Gas mass” mode): a flask is weighed with air and with the unknown gas, and its volume is measured with water; $m (air) = ρ_{air} V$ , $m (gas) = m_{2} - m_{1} + m (air)$ , $M = 29 \cdot m (gas) / m (air)$ . Reducing a gas volume to STP: $V_{0} = V \cdot \frac{p}{101.325} \cdot \frac{273.15}{T}$ .

Equivalent and the law of equivalents (laboratory)

An equivalent is the amount of a substance that combines with, or replaces, 1 mol of hydrogen atoms (1.008 g of H) or 8 g of oxygen. The equivalent mass of a metal is $E = A / valency$ : $E (Mg) = 24.305 : 2 = 12.15$ g/mol, $E (Al) = 26.98 : 3 = 8.99$ g/mol. The law of equivalents: substances react in masses proportional to their equivalents, $m_{1} / m_{2} = E_{1} / E_{2}$ . 1 mol-eq of H₂ (1.008 g) is 11.2 L at STP. Experiment: a metal is dissolved in hydrochloric acid and the H₂ is collected under water in a eudiometer. The gas is saturated with water vapour, so the partial pressure of hydrogen is $p (H_{2}) = p - p (H_{2} O)$ (at 20 °C $p (H_{2} O) = 2.34$ kPa). Then $V_{0}$ and $E = m \cdot 11 200 / V_{0}$ . In redox reactions the equivalent is determined by the number of electrons given or taken: $E = M / z$ (for example, for KMnO₄ in an acidic medium $158.0 : 5 = 31.6$ g/mol).

Safety and the model

Safety: work with hydrochloric acid in safety goggles and gloves; if acid gets in the eyes or on the skin, rinse with plenty of water. Hydrogen forms an explosive mixture with air — do not light a flame nearby. Work with gas cylinders is demonstrated by the teacher. Model: gases are taken to be ideal (the error at room conditions is less than 1 %); the mass of the balloon rubber is not taken into account; the metal dissolves completely, the acid is in excess; the water vapour pressure uses Buck's (1996) formula; densities are from the CRC handbook (20 °C). Tashkent is about 450–500 m above sea level, so a barometer here usually reads less than 101.3 kPa — about 95–96 kPa; in the measurement take the real pressure in the room.

Did you know?

Since 2019 the Avogadro constant has been fixed as an exact number: Nₐ = 6.02214076·10²³ mol⁻¹. The mole is defined through this number, and the kilogram through the Planck constant.
Amedeo Avogadro published his hypothesis in 1811, but it was the French physicist Jean Perrin who proposed in 1909 that the constant be named after him. Perrin received the Nobel Prize in 1926 for measuring it in experiments on Brownian motion and thereby proving that matter is made of atoms.

Inside the experiment

The experiment starts from a real-life situation: you predict the result, change the controls, run the experiment and draw a conclusion. Inside are missions, theory, a quiz and a guided tour in which the camera moves and explains the key points.

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