Multiple Proportions: Basic Laws of Chemistry - Chemistry (Grade 10)

Two Elements Can Make Several Compounds

The law of multiple proportions applies when the same two elements form more than one compound. If the mass of one element is kept fixed, the masses of the other element that combine with it form a simple whole-number ratio.

OpenStax Chemistry: Atoms First 2e explains this law as a fixed mass of one element reacting with masses of another element in a small whole-number ratio in openstax.org. Britannica summarizes the same mass pattern in britannica.com.

So this law does not compare any two substances at random. The compounds must be different, but their elements must be the same.

Keep One Ratio Base Fixed

Suppose two compounds are both made from elements $\mathrm{A}$ and $\mathrm{B}$ . To read Dalton's law, first hold the mass of $\mathrm{A}$ fixed, then compare the mass of $\mathrm{B}$ that combines with it.

\frac{m_{\mathrm{B\ in\ compound\ 2}}}{m_{\mathrm{B\ in\ compound\ 1}}} = \text{simple whole-number ratio}

The fixed mass is the important move. If both element masses change at once, the pattern can look messy. Once one element is put on the same basis, the multiple ratio becomes visible.

Choose a compound pair below. The model shows the element held fixed and the element whose mass changes.

Ratio Pairing

Compare two compounds made from the same elements. One element mass is held fixed, then the other is read as a simple ratio.

With the same mass of $\mathrm{H}$ , $\mathrm{H_2O_2}$ carries twice the mass of $\mathrm{O}$ as $\mathrm{H_2O}$ .

Mass held fixed: The hydrogen mass is fixed at $2\ \mathrm{g}$ .
Changing ratio: The oxygen masses are $16:32 = 1:2$ .

In each pair above, one element is used as the comparison base. Once that base is the same, the element whose count changes is read as a simple mass ratio.

Water and Hydrogen Peroxide

Water has the formula $\mathrm{H_2O}$ , while hydrogen peroxide has the formula $\mathrm{H_2O_2}$ . PubChem records those formulas in its pubchem.ncbi.nlm.nih.gov and pubchem.ncbi.nlm.nih.gov.

Using $A_r\mathrm{(H)} = 1$ and $A_r\mathrm{(O)} = 16$ , the hydrogen mass in the two formulas can be made the same, $2\ \mathrm{g}$ .

Compound	Mass of $\mathrm{H}$	Mass of $\mathrm{O}$
$\mathrm{H_2O}$	$2\ \mathrm{g}$	$16\ \mathrm{g}$
$\mathrm{H_2O_2}$	$2\ \mathrm{g}$	$32\ \mathrm{g}$

Because the hydrogen mass is already the same, compare the oxygen masses directly.

m_{\mathrm{O\ in\ H_2O}} : m_{\mathrm{O\ in\ H_2O_2}} = 16:32 = 1:2

The ratio $1:2$ is a simple whole-number ratio. That means these data fit the law of multiple proportions.

Nitrogen Oxides

Nitrogen and oxygen can form several compounds. Two examples are dinitrogen monoxide, or nitrous oxide, $\mathrm{N_2O}$ , and nitrogen dioxide, $\mathrm{NO_2}$ . PubChem records $\mathrm{N_2O}$ and $\mathrm{NO_2}$ in its pubchem.ncbi.nlm.nih.gov and pubchem.ncbi.nlm.nih.gov.

To make the oxygen mass the same, compare $2\mathrm{N_2O}$ with $\mathrm{NO_2}$ . Both contain $32\ \mathrm{g}$ of oxygen.

Compared compound	Mass of $\mathrm{O}$	Mass of $\mathrm{N}$
$2\mathrm{N_2O}$	$32\ \mathrm{g}$	$56\ \mathrm{g}$
$\mathrm{NO_2}$	$32\ \mathrm{g}$	$14\ \mathrm{g}$

Now that the oxygen mass is the same, compare the nitrogen masses:

m_{\mathrm{N\ in\ 2N_2O}} : m_{\mathrm{N\ in\ NO_2}} = 56:14 = 4:1

Multiplying $\mathrm{N_2O}$ to $2\mathrm{N_2O}$ is only a way to put oxygen on the same mass basis. The compound formula is still $\mathrm{N_2O}$ , not a new compound.

Mistakes to Avoid

Do not compare total compound masses. Compare the mass of one element after the other element has been held fixed.
Do not use this law for two substances made from different elements. It only applies to compound pairs made from the same elements.

The law of constant composition reads one compound. The law of multiple proportions reads several compounds made from the same element pair. That small shift is why Dalton's law became strong evidence that atoms combine in whole-number counts.