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The fugacities of water and carbon dioxide are calculated as a function of temperature for a closed container, which is a model of a can of soda. The concentrations of the two components are calculated in both the liquid and gas phases. As temperature increases, the pressure increases, and therefore the fugacities increase. Note that the CO₂ concentration is much lower than the H₂O concentration in the liquid phase, but the CO₂ concentration is much higher than the H₂O concentration in the gas phase. Because the gas phase is assumed to be ideal, the fugacities of CO₂ and H₂O in both phases are equal to their gas-phase partial pressures, and thus the CO₂ fugacity is much higher than the H₂O fugacity. As the temperature increases, the CO₂ concentration in liquid water decreases. However, as the temperature increases the pressure increases, and a higher CO₂ pressure increases the CO₂ concentration in water, so the net effect is that CO₂ concentration in the liquid phase does not change much as the temperature increases.

Details

The fugacity of CO₂ dissolved in water was calculated using Henry's law and freezing point depression measurements:

where \(\Delta T_{fp}\) is the difference between the freezing point of water and the freezing point of carbon dioxide in water (\(T_{\mathrm{H_2O}}\) and \(T_{sol}\)) (K), \(k_{fp}\) is the freezing point depression constant for water ([°C kg]/mol), \(m_{\mathrm{CO_2}}\) is the molality of CO₂, \(k_{H}\) is Henry's law constant (kg/[mol bar]), \(k_{H}^{o}\) is Henry's constant at 273 K, \(\frac{\delta\,\ln(k_{H})}{\delta\,T^{-1}}=2400\) is a constant, \(T\) is temperature (K), \(P_{\mathrm{CO_2}}\) is the partial pressure of CO₂ (bar), and \(f_{\mathrm{CO_2}}\) is the fugacity of CO₂ (bar).

The fugacity of water was calculated from the saturation pressure of water \(P_{\mathrm{H_2O}}^{\,sat}\) using the Antoine equation:

where \(f_{\mathrm{H_2O}}\) is the fugacity of water (bar).

Reference
[1] T. S. Kuntzleman and C. Richards, "Another Method for Determining the Pressure inside an Intact Carbonated Beverage Can (or Bottle)," Journal of Chemical Education, 87(9), 2010 p. 993. doi:10.1021/ed100255g.

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