Gas molecules travel in random paths and collide with one another and the walls of their container. These collisions exert a pressure per unit area and also cause the gases to occupy a volume. Both the pressure and volume are affected by temperature. The interrelationships between these three variables were formulated by Boyle, Charles, and Gay-Lussac, and can be applied to pharmaceutical aerosols.
Boyle’s Law states that:
when temperature does not change
Charles’s or Gay-Lussac’s Law states that:
when pressure does not change
If two sets of conditions are being considered, these equations can be combined to obtain the relationship:
where the subscripts 1 and 2 refer to two different conditions. Although the P, V, and T of each condition may be different, the ratio is constant and can be mathematically expressed as:
This equation is derived considering only 1 mole (i.e., one gram molecular weight) of ideal gas. If n moles of gas were to be considered, it becomes:
which is known as the General Ideal Gas Law. R is the molar gas constant and is used with many different units depending on the mathematical application: 8.314 J/K/mole, 0.08205 L atm/K/mole, and 1.987 cal/K/mole.
The relevance of the gas laws to pharmaceutical aerosols can be seen in the following examples.
Example 1: What is the weight of nitrogen in an 8 fluid ounces aerosol container filled with 6 fluid ounces of viscous ointment? The container is pressurized to 90 psig and the temperature is 25°C.
Determine the following values:
From a rearrangement of the General Ideal Gas Law:
Converting the number of moles to a weight can be done using the atomic weight of nitrogen (i.e., 14) and its valence, which is 2:
Example 2: If 3 fluid ounces of the aerosol are dispensed, what is the resulting pressure in the container?
The Ideal Gas Law can be rearranged, noting that T1 = T2 (i.e., temperature has not changed), by setting V1 and P1 to the initial values for the container (i.e., 2 fluid ounces of gases), and solving for V2 and P2 after the container as emitted 3 fluid ounces (i.e., there are now 5 fluid ounces of gas):
Example: What is the pressure remaining in the container when all of the product has been dispensed?
The equation from the previous example can still be used. However, the gas now occupies 8 fluid ounces.
A liquefied gas propellant can be considered as a solution. Molecules of the solution will have escape tendencies that will create a vapor pressure above the solution. According to Raoult’s Law, if a solute is added to a solvent, the solvent vapor pressure will be decreased proportional to the mole fraction of the solute added. If the added solute has an appreciable vapor pressure itself, its vapor pressure will also be decreased as the result of its dilution in the solvent. The total vapor pressure of a mixture of propellants can be determined as the sum of the partial pressures of each component (Dalton’s Law). The partial pressure of a component can be determined as the mole fraction of the component multiplied by the vapor pressure of the pure compound. If propellant a and b are mixed, the partial pressures can be calculated as:
The total vapor pressure of the aerosol would be the sum of the two partial pressures calculated above:
Example: What is the vapor pressure of a mixture of propellants 11 and 12 in a 70 g to 30 g ratio?
apsia (pounds per square inch absolute) = psig (pounds per square inch gauge + 14.7)
||Vapor Pressure 70°F (psia)a
Determine the mole fractions of each propellant.
Determine the partial pressure for each propellant.
The total vapor pressure for the propellant mixture will be: