Multiple Proportions

Two Elements Can Make Several Compounds

The law of multiple proportions applies when the same two elements form more than one compound. Hold the mass of one element fixed, then compare the corresponding masses of the other element. Those masses form a small whole-number ratio.

Fix the Mass of One Element for Comparison

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.

For the same mass of element A, let $m_{B,1}$ and $m_{B,2}$ denote the masses of B in compounds 1 and 2. Their ratio $m_{B,2}/m_{B,1}$ is a simple whole-number ratio.

The mass of one element must be the same in both compounds before the other element's masses are compared. If both element masses change at once, the calculation cannot directly test whether the second element's mass ratio is a simple whole number.

Compare Compounds Made from the Same Elements

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

Comparing Masses in Two Compounds

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 $\mathrm{H}$ , $\mathrm{H_2O_2}$ contains 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 $\mathrm{H_2O_2}$ .

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.

\begin{aligned} &m_{\mathrm{O\ in\ H_2O}} : m_{\mathrm{O\ in\ H_2O_2}} = 16:32 \\ &= 1:2 \end{aligned}

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, $\mathrm{NO_2}$ .

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:

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

Multiplying $\mathrm{N_2O}$ to $2\mathrm{N_2O}$ puts oxygen on the same mass basis. The compound formula stays $\mathrm{N_2O}$ .

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.

For one compound, the law of constant composition compares the mass ratio of its elements. For two or more compounds made from the same element pair, the law of multiple proportions compares those ratios. The resulting small whole-number ratios supported Dalton's claim that atoms combine in whole-number counts.