The Gaseous State
We have a pretty good understanding of the gaseous state in terms of the link between the variables describing the microscopic (atomic-scale) realm, and the variables describing the macroscopic realm.
In the microscopic world we can talk about position, velocity, and mass of the individual atoms, all of which are governed by Newton's laws of motion. By the use of statistical mechanics and averages we can then relate these variables to the macroscopic world by talking about such quantities as pressure, volume, temperature, and moles. For now, though, let's focus on the macroscopic world and search for an equation of state that will tell us the relationships between all these macroscopic world quantities.
Boyle's Law
We start with the experimental observations of Robert Boyle, who showed that the quantities of pressure times volume are a constant. This is called Boyle's Law. This is shown in the diagrams below.
Each line on the graph is a line of constant temperature, and is called an isotherm. In a plot of P versus 1/V, we see that the isotherms are straight lines with constant slopes.
For an ideal gas the pressure times the volume is constant, but for real-world gases this is not always true.
The straight horizontal line in the P V versus P graph is for an ideal gas, while the other two lines are those of some real gases. The thing to notice here is that as pressure goes to zero all gases become ideal, and this is shown by the fact that as P goes to zero, all lines converge to the same point.
Using these relationships, we can talk about the topic of compressibility and one of the hazards of scuba diving. In the graph below we see that a low compressibility means that something is hard to squeeze, i.e., a big change in pressure gives a small change in volume. Likewise, a high compressibility means that something is easier to squeeze, i.e., a small change in pressure gives a large change in volume.
You can think of compressibility in terms of density. Something that has a high density, like a solid, has a very low compressibility since it takes a lot of pressure to change the volume even slightly, whereas something that has a low density, such as a gas, has a very high compressibility.
Now, let's look at how these ideas relate to diving. It is well known among divers that diving at the surface can be more dangerous than deeper diving. We can understand this by first noting that for every 10 meters you descend in the water, the pressure increases by about 1 atm.
Therefore, when you hold your breath, you create a closed system with your lungs and thus Boyle's law will hold. If you are down at 90 meters (at 10 atm) and you rise 10 meters to 80 meters (at 9 atm), the pressure has decreased by about 10%, and since PV is a constant your lungs expand by about 10% (probably not too bad). Now if you are at 10 meters (at 2 atm), and you rise 10 meters to the surface (at 1 atm) the pressure had decreased by 50% and expanding your lungs by this factor could cause significant damage, maybe death!
Charles' Law
Now let's look at the relationships between volume and temperature. These relationships were discovered by Jacque Charles and Joseph Louie Gay-Lussac.
In the plot of V-vs-T above, we observe that the relationship is linear, with an intercept of absolute zero on the Kelvin scale. The slope depends on the pressure and the amount of gas. The dashed lines on the graph are to represent the fact that you can not achieve absolute zero.
Now if we put Boyle's and Charles' laws together, we can come up with the equation of state that I spoke of earlier. Thus we know...
- if Temperature is constant, then PV=constant.
- if Pressure is constant, then Volume is proportional to Temperature.
- if Pressure is constant, then Volume is proportional to n (the amount of gas in moles).
Therefore, we can write
P V = n R T
R is a proportionality constant called the universal gas constant and has the value R=8.314 J/K-mole (Units are important!).
Making the definitions for standard temperature and pressure...
Standard Temperature: T = 273 K
Standard Pressure: P = 1 atm
we find using P V = n R T that 1 mole at STP (Standard Temperature and Pressure) has a volume of 22.4 liters.
2.96g Mercuric Chloride vaporized in a 1.0L bulb at 680 K and P=453 torr. What is the molar mass?
Assuming an ideal gas, i.e., P V = n R T
- We know everything except n,
therefore we can solve for n (number
of moles).
- Molar Mass = 2.69g / 0.011 moles = 277 g/moles.
We can also use the ideal gas law as a way of converting volumes into moles when calculating stoichiometric quantities in chemical reactions.
A sample of CH4(g) having a volume of 2.80 Liters at 25°C and 1.65 atm, was ignited with a sample of oxygen gas having a volume of 35.0 Liters at 31°C and 1.25 atm to produce CO2 and H2O vapor. Calculate the volume of CO2 formed at a pressure of 2.50 atm and a temperature of 125°C.
- First find the limiting reagent.
(For solids, liquids and gases—we can always calculate the number of moles from the mass and the molecular weight, for an ideal gas, however, we can also convert volume to moles using PV=nRT).
- Volume of CO2 produced is:
So there are 2.47 liters of CO2 produced.