In chemistry, "partial pressure" means the pressure that each gas present in a mixture exerts on the container, for example a flask, a diver's air cylinder or the limits of an atmosphere; it is possible to calculate it if you know the quantity of each gas, the volume it occupies and its temperature. You can also add up the various partial pressures and find the total one exerted by the mixture; alternatively, you can first calculate the total and get the partial values.
Steps
Part 1 of 3: Understanding the Properties of Gases
Step 1. Treat each gas as if it were "perfect"
In chemistry, an ideal gas interacts with others without being attracted to their molecules. Each molecule collides and bounces on the others like a billiard ball without deforming in any way.
- The pressure of an ideal gas increases as it is compressed into a smaller vessel and decreases as the gas expands into larger spaces. This relationship is called Boyle's law, after its discoverer Robert Boyle. Mathematically it is expressed with the formula k = P x V or more simply k = PV, where k is the constant, P is the pressure and V the volume.
- Pressure can be expressed in many different units of measurement, such as the pascal (Pa) which is defined as the force of a newton applied on a surface of one square meter. Alternatively, the atmosphere (atm), the pressure of the earth's atmosphere at sea level, can be used. One atmosphere is equivalent to 101, 325 Pa.
- The temperature of ideal gases rises as their volume increases and falls when the volume decreases; this relation is called Charles' law and was enunciated by Jacques Charles. It is expressed in mathematical form as k = V / T, where k is a constant, V is the volume and T the temperature.
- The temperatures of the gases taken into consideration in this equation are expressed in degrees kelvin; 0 ° C corresponds to 273 K.
- The two laws described so far can be combined together to obtain the equation k = PV / T which can be rewritten: PV = kT.
Step 2. Define the units of measurement in which the quantities of gases are expressed
The substances in the gaseous state have both a mass and a volume; the latter is generally measured in liters (l), while there are two types of masses.
- Conventional mass is measured in grams or, if the value is large enough, in kilograms.
- Since gases are typically very light, they are also measured in other ways, by molecular or molar mass. The molar mass is defined as the sum of the atomic mass of each atom present in the compound that generates the gas; atomic mass is expressed in the unified atomic mass unit (u), which is equal to 1/12 of the mass of a single carbon-12 atom.
- Since atoms and molecules are too small entities to work with, the amount of gas is measured in moles. To find the number of moles present in a given gas, the mass is divided by the molar mass and represented by the letter n.
- You can arbitrarily replace the constant k in the gas equation with the product of n (the number of moles) and a new constant R; at this point, the formula takes the form of: nR = PV / T or PV = nRT.
- The value of R depends on the unit used to measure the pressure, volume and temperature of the gases. If the volume is defined in liters, the temperature in kelvins and the pressure in atmospheres, R is equal to 0.0821 l * atm / Kmol, which can be written as 0.0821 l * atm K-1 mol -1 to avoid using the division symbol in the unit of measure.
Step 3. Understand Dalton's law for partial pressures
This statement was elaborated by the chemist and physicist John Dalton, who first advanced the concept that chemical elements are composed of atoms. The law states that the total pressure of a gas mixture is equal to the sum of the partial pressures of each gas that makes up the mixture itself.
- The law can be written in mathematical language such as Ptotal = P1 + P2 + P3… With a number of addends equal to that of the gases making up the mixture.
- Dalton's law can be expanded when working with gas of unknown partial pressure but with known temperature and volume. The partial pressure of a gas is the same as it would have if it were present alone in the vessel.
- For each partial pressure, you can rewrite the perfect gas equation to isolate the P term of the pressure to the left of the equality sign. So, starting from PV = nRT, you can divide both terms by V and get: PV / V = nRT / V; the two variables V on the left cancel each other out leaving: P = nRT / V.
- At this point, for each variable P in Dalton's law you can substitute the equation for the partial pressure: P.total = (nRT / V) 1 + (nRT / V) 2 + (nRT / V) 3…
Part 2 of 3: Calculate Partial Pressures First, then Total Pressures
Step 1. Define the partial pressure equation of the gases under consideration
As an example, suppose you have three gases contained in a 2-liter flask: nitrogen (N.2), oxygen (O2) and carbon dioxide (CO2). Each quantity of gas weighs 10 g and the temperature is 37 ° C. You have to find the partial pressure of each gas and the total pressure exerted by the mixture on the walls of the container.
- The equation is therefore: P.total = Pnitrogen + Poxygen + Pcarbon dioxide.
- Since you want to find the partial pressure exerted by each gas, knowing the volume and the temperature, you can calculate the amount of moles thanks to the mass data and rewrite the equation as: Ptotal = (nRT / V) nitrogen + (nRT / V) oxygen + (nRT / V) carbon dioxide.
Step 2. Convert the temperature to kelvins
Those provided by the statement are expressed in degrees Celsius (37 ° C), so just add the value 273 and you get 310 K.
Step 3. Find the number of moles for each gas that makes up the mixture
The number of moles is equal to the mass of the gas divided by its molar mass, which in turn is the sum of the atomic masses of each atom in the compound.
- For the first gas, nitrogen (N.2), each atom has a mass of 14. Since nitrogen is diatomic (it forms molecules with two atoms), you have to multiply the mass by 2; consequently, the nitrogen present in the sample has a molar mass of 28. Divide this value by the mass in grams, 10 g, and you get the number of moles which corresponds to approximately 0.4 mol of nitrogen.
- For the second gas, oxygen (O2), each atom has an atomic mass equal to 16. This element also forms diatomic molecules, so you have to double the mass (32) to obtain the molar mass of the sample. By dividing 10 g by 32 you come to the conclusion that there is about 0.3 mol of oxygen in the mixture.
- The third gas, carbon dioxide (CO2), is composed of three atoms: one of carbon (atomic mass equal to 12) and two of oxygen (atomic mass of each equal to 16). You can add the three values (12 + 16 + 16 = 44) to know the molar mass; divide 10 g by 44 and you get about 0.2 mol of carbon dioxide.
Step 4. Replace the equation variables with moles, temperature, and volume information
The formula should look like this: Ptotal = (0.4 * R * 310/2) nitrogen + (0.3 * R * 310/2) oxygen + (0, 2 * R * 310/2) carbon dioxide.
For reasons of simplicity, the units of measurement have not been inserted next to the values, as they are canceled by carrying out the arithmetic operations, leaving only the one associated with the pressure
Step 5. Enter the value for the constant R
Since partial and total pressure are reported in atmospheres, you can use the number 0.0821 l * atm / K mol; substituting it for the constant R you get: Ptotal =(0, 4 * 0, 0821 * 310/2) nitrogen + (0, 3 * 0, 0821 * 310/2) oxygen + (0, 2 * 0, 0821 * 310/2) carbon dioxide.
Step 6. Calculate the partial pressure of each gas
Now that all the known numbers are in place, you can do the math.
- As for nitrogen, multiply 0, 4 mol by the constant of 0, 0821 and the temperature equal to 310 K. Divide the product by 2 liters: 0, 4 * 0, 0821 * 310/2 = 5, 09 atm approximately.
- For oxygen, multiply 0.3 mol by the constant of 0.0821 and the temperature of 310 K, and then divide it by 2 liters: 0.3 * 0.3821 * 310/2 = 3.82 atm approximately.
- Finally, by carbon dioxide multiply 0.2 mol by the constant of 0.0821, the temperature of 310 K and divide by 2 liters: 0.2 * 0.0821 * 310/2 = approximately 2.54 atm.
- Add all the addends to find the total pressure: P.total = 5, 09 + 3, 82 + 2, 54 = 11, 45 atm approximately.
Part 3 of 3: Calculate Total Pressure then Partial Pressure
Step 1. Write the partial pressure formula as above
Again, consider a 2-liter flask that contains three gases: nitrogen (N.2), oxygen (O2) and carbon dioxide. The mass of each gas is equal to 10 g and the temperature in the container is 37 ° C.
- The temperature in degrees kelvin is 310 K, while the moles of each gas are approximately 0.4 mol for nitrogen, 0.3 mol for oxygen and 0.2 mol for carbon dioxide.
- As for the example in the previous section, it indicates the pressure values in atmospheres, for which you must use the constant R equal to 0, 021 l * atm / K mol.
- Consequently, the equation of partial pressures is: P.total =(0, 4 * 0, 0821 * 310/2) nitrogen + (0, 3 * 0, 0821 * 310/2) oxygen + (0, 2 * 0, 0821 * 310/2) carbon dioxide.
Step 2. Add the moles of each gas in the sample and find the total number of moles of the mixture
Since the volume and temperature do not change, not to mention the fact that the moles are all multiplied by a constant, you can take advantage of the distributive property of the sum and rewrite the equation as: Ptotal = (0, 4 + 0, 3 + 0, 2) * 0, 0821 * 310/2.
Do the sum: 0, 4 + 0, 3 + 0, 2 = 0, 9 mol of the gas mixture; in this way, the formula is simplified even more and becomes: Ptotal = 0, 9 * 0, 0821 * 310/2.
Step 3. Find the total pressure of the gas mixture
Do the multiplication: 0, 9 * 0, 0821 * 310/2 = 11, 45 mol or so.
Step 4. Find the proportions of each gas to the mixture
To proceed, simply divide the number of moles of each component by the total number.
- There are 0.4 moles of nitrogen, so 0.4/0.7 = 0.44 (44%) approximately;
- There are 0.3 mol of oxygen, so 0.3/0.9 = 0.33 (33%) approximately;
- There are 0.2 moles of carbon dioxide, so 0.2/0.9 = 0.22 (22%) approximately.
- Although adding the proportions gives a total of 0.99, in reality the decimal digits repeat themselves periodically and by definition you can round the total to 1 or 100%.
Step 5. Multiply the percentage amount of each gas by the total pressure to find the partial pressure:
- 0.44 * 11.45 = 5.04 atm approximately;
- 0.33 * 11.45 = 3.78 atm approximately;
- 0, 22 * 11, 45 = 2, 52 atm approx.
