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Example 2.2: Laws of Definite and Multiple Proportions

A sample of compound A (a clear, colorless gas) is analyzed and found to contain 4.27 g carbon and 5.69 g oxygen. A sample of compound B (also a clear, colorless gas) is analyzed and found to contain 5.19 g carbon and 13.84 g oxygen. Are these data an example of the law of definite proportions, the law of multiple proportions, or neither? What do these data tell you about substances A and B?

Solution

In compound A, the mass ratio of carbon to oxygen is:

\(m_{\mathrm{\ce{C},compA}}\) \(= 4.27\ \mathrm{g}\)


\(m_{\mathrm{\ce{O},compA}}\) \(= 5.69\ \mathrm{g}\)


\(r_{\mathrm{\ce{C:O},compA}}\) \(= \dfrac{m_{\mathrm{\ce{C},compA}}}{m_{\mathrm{\ce{O},compA}}}\)

\(\ \ \ =\dfrac{4.27\ \mathrm{g}}{5.69\ \mathrm{g}}\)

\(\ \ \ =0.750\)


In compound B, the mass ratio of carbon to oxygen is:

\(m_{\mathrm{\ce{C},compB}}\) \(= 5.19\ \mathrm{g}\)


\(m_{\mathrm{\ce{O},compB}}\) \(= 13.84\ \mathrm{g}\)


\(r_{\mathrm{\ce{C:O},compB}}\) \(= \dfrac{m_{\mathrm{\ce{C},compB}}}{m_{\mathrm{\ce{O},compB}}}\)

\(\ \ \ =\dfrac{5.19\ \mathrm{g}}{13.84\ \mathrm{g}}\)

\(\ \ \ =0.3750\)


The ratios are different, so this is not the law of definite proportions (and compound A is different from compound B because it has different composition).
The ratio of these ratios is:

\(r_{\mathrm{r}}\) \(= \dfrac{r_{\mathrm{\ce{C:O},compA}}}{r_{\mathrm{\ce{C:O},compB}}}\)

\(\ \ \ =\dfrac{0.750}{0.3750}\)

\(\ \ \ =2.001\)


This supports the law of multiple proportions. This means that A and B are different compounds, with A having twice as much carbon per amount of oxygen (or twice as much oxygen per amount of carbon) as B. A possible pair of compounds that would fit this relationship would be compA \( = \) \(\ce{CO}\) and compB \( = \) \(\ce{CO2}\).