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Why does an open soda go flat according to the principles of the Ideal Gas Law?
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7. Gases
The Ideal Gas Law
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The density of an ideal gas is directly proportional to which of the following quantities?
A
Molar mass (M)
B
Gas constant (R)
C
Temperature (T)
D
Pressure (P)
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Recall the ideal gas law: \(\displaystyle PV = nRT\), where \(P\) is pressure, \(V\) is volume, \(n\) is moles, \(R\) is the gas constant, and \(T\) is temperature.
Express the number of moles \(n\) in terms of mass \(m\) and molar mass \(M\): \(\displaystyle n = \frac{m}{M}\).
Rewrite the ideal gas law substituting \(n\): \(\displaystyle PV = \frac{m}{M}RT\).
Rearrange to solve for density \(\rho = \frac{m}{V}\): \(\displaystyle P = \frac{\rho}{M}RT \implies \rho = \frac{PM}{RT}\).
From the equation \(\displaystyle \rho = \frac{PM}{RT}\), observe that density \(\rho\) is directly proportional to pressure \(P\) and molar mass \(M\), and inversely proportional to temperature \(T\). The gas constant \(R\) is a constant and does not affect proportionality.
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