Example 9.9: Using the Ideal Gas Law

Methane, \(\ce{CH4}\), is being considered for use as an alternative automotive fuel to replace gasoline. One gallon of gasoline could be replaced by 655 g of \(\ce{CH4}\). What is the volume of this much methane at 25 °C and 745 torr?

Solution

\(V_{\mathrm{\ce{CH4}}}\) = ?


\(m_{\mathrm{\ce{CH4}}}\) \(= 655.\ \mathrm{g}\)


\(T\) \(= 25.\ \mathrm{°aC}\)


\(P\) \(= 745.\ \mathrm{torr}\)


\(P \cdot V = n \cdot R \cdot T\)     

\(R\) \(= 8.314\ \frac{\mathrm{J}}{\mathrm{mol}\ \mathrm{K}}\)


\(n_{\mathrm{\ce{CH4}}}\) \(= \dfrac{m_{\mathrm{\ce{CH4}}}}{M_{\mathrm{\ce{CH4}}}}\)

\(\ \ \ =\dfrac{655.\ \mathrm{g}}{16.043\ \frac{\mathrm{g}}{\mathrm{mol}}}\)

\(\ \ \ =40.83\ \mathrm{mol}\)


\(V_{\mathrm{\ce{CH4}}}\) \(= \dfrac{n_{\mathrm{\ce{CH4}}} \cdot R \cdot T}{P}\)

\(\ \ \ =\dfrac{40.83\ \mathrm{mol} \cdot 8.314\ \frac{\mathrm{J}}{\mathrm{mol}\ \mathrm{K}} \cdot 25.\ \mathrm{°aC}}{745.\ \mathrm{torr}}\)

\(\ \ \ =\dfrac{339.44\ \frac{\mathrm{J}}{\mathrm{K}} \cdot 298.\ \mathrm{K}}{745.\ \mathrm{torr}}\)

\(\ \ \ =\dfrac{1.0120\times 10^{5}\ \mathrm{J}}{745.\ \mathrm{torr}}\)

\(\ \ \ =1.019\ \mathrm{m}^{3}\)


It would require 1020 L (269 gal) of gaseous methane at about 1 atm of pressure to replace 1 gal of gasoline. It requires a large container to hold enough methane at 1 atm to replace several gallons of gasoline.
\(V_{\mathrm{\ce{CH4}}}\) \(= 1.019\ \mathrm{m}^{3}\)

\(\ \ \ =1019.\ \mathrm{L}\)