Example 9.11: Derivation of a Density Formula from the Ideal Gas Law
Use
\(\ce{PV}\) \( = \mathrm{nRT}\) to derive a formula for the density of gas in g/L
Solution
\(P \cdot V_{\mathrm{gas}} = n_{\mathrm{gas}} \cdot R \cdot T\)
The density is defined as mass per volume. To get mass into the equation, multiply each side by the molar mass of the gas:
\(M_{\mathrm{gas}} \cdot P \cdot V_{\mathrm{gas}} = m_{\mathrm{gas}} \cdot R \cdot T\)
To have
\(m_{\mathrm{gas}}\) /
\(V_{\mathrm{gas}}\) on the right hand side, divide by
\(V_{\mathrm{gas}}\), R, and T:
\(\dfrac{M_{\mathrm{gas}} \cdot P}{R \cdot T} = \dfrac{m_{\mathrm{gas}}}{V_{\mathrm{gas}}}\)
Finally, we can substiture ρ, the symbol for density to arrive at
\(ρ = \dfrac{m_{\mathrm{gas}}}{V_{\mathrm{gas}}} = \dfrac{M_{\mathrm{gas}} \cdot P}{R \cdot T}\)