You might have wondered why we only need to specify two of the three state variables and why we picked temperature and pressure. Which of the two we pick is arbitrary and we can pick any two; the two that are easiest to measure are usually good choices. I could pick temperature and volume, or temperature and pressure, or pressure and volume. This is because once two of the three state variables are specified, we will have no choice in the value of third variable. It will be set by the values of the other two state variables. To see why this is so we need one more definition: the equation of state is a mathematical expression that links the state variables. Because they have the weakest intermolecular interactions, gases have a particularly simple equation of state that is accurate for wide ranges of temperature, pressure and volume. This equation of state is called the ideal gas law

PV = nRT

where n is the number of moles of gas, R is the gas constant and T, P and V are the state variables. For this equation to work, your temperature needs to be expressed in Kelvin (See conversions and definitions of temperature units.). When we assume that the equation of state for a gas can be accurately described by the ideal gas law, we say the gas behaves ideally. For an ideal gas, you can use an on-line ideal gas calculator to find the last state variable once you have the other two, or you can calculate it yourself by picking R in the correct units and doing the algebra. Make sure your units of R match the units of your pressure and volume.

If you want some practice solving problems with the ideal gas law, try problems; there are solutions at this web site too. One hint: STP stands for standard temperature and pressure where T = 273.15 K and 1.00 atmospheres (101.3 kPa). If you are looking for some interesting high school science projects to do to illustrate the use of the ideal gas law you can try a dry-lab on air bags or a wet lab on the reaction of magnesium with water (careful! the gas produced is H₂. Your school might not be equipped to handle this sort of a lab, even as a demo). This latter site also has a teachers' note page.

A slightly more accurate - and thus more complicated- equation of state for gases is that given by the Van der Waals formula. We might need to use this equation of state in place of the ideal gas law when the interactions between gas molecules are relatively large, for example when we are close to the point that the gas will condense to form a solid or a liquid. A very simple, but not too accurate, equation of state for solids and liquids is that the volume is independent of temperature and pressure: V = constant. These equations of state are our attempts to model the behavior of chemical systems mathematically. Some approximations are good under some circumstances, but not so good under others that span bigger ranges of T, P or that demand that you know the relationship between T, P and V very precisely. But remember the system always knows what it wants to do. If you set the temperature and pressure of any closed system, say a balloon filled with helium, there will be one and only one volume that the helium can adopt at equilibrium. Similarly, if you set the pressure and volume of, say, a bar of gold you also set its temperature.

We end this part of the thermo course with one last set of definitions. Sometimes having two state variables is too big of a complication and we want only one value to vary at a time. We can do this by holding one variable constant, measuring another and not worrying about the third (which is set by the values of the other two). Sometimes nature is kind and even does this for us. For example, if you carry out a chemical reaction on the top of a laboratory bench, the pressure of the system stays essentially constant because the pressure of the laboratory doesn't change perceptibly during the reaction. Under these conditions, P = constant, and we can measure the system temperature T and be done with it. (Yes, we could have picked V instead, but it turns out T will be much more convenient.) We only have a single variable needed to specify the state of our system. These kinds of changes are called isobaric (iso = same; bar = pressure) and reactions carried out under constant pressure are said to be performed isobarically.

Changes carried out at constant volume are called isochors - the adjective is isochoric and the adverb isochorically. Changes involving only condensed phases (solids and liquids) can often be approximated as occurring isochorically, as can reactions involving only gases held in rigid containers. When the temperature is held constant, the change is carried out isothermally and we can approximate this situation by immersing our system in a thermostatted water bath. A final thermodynamic path for change is one that is adiabatic where we do not let our system exchange heat with the surroundings. While this term perhaps is less familiar than the others describing thermodynamic pathways, it is one of the most important and we will use it often. We actually approximate adiabatic conditions all the time in our everyday experiences. Hot coffee placed in a styrofoam cup exchanges heat only very slowly with the air around it. Soup placed in a thermos at 8 a.m. is still hot at lunch time. Meanwhile, the egg salad sandwiches stored on ice in the cooler are still chilled and (we hope!) salmonella-free. A well-insulated system will be called adiabatic because it cannot exchange heat with the surroundings due to its thermal isolation or does so very slowly so that we can ignore the heat exchange over the time of our experiment.

Because thermodynamics is a very quantitative field, it is easiest to do and is best understood if we use a fair bit of mathematics. One of the content integration topics for this course module is calculus and you will understand thermodynamics best if you take some time to learn the calculus behind it. However, most high school texts do not require more mathematical sophistication than is involved in using simple algebra. Also, you need to remember the dynamics part of thermodynamics. Things will be changing! We will start out in one state of the system (one set of values for temperature, pressure and volume) and let the system change until it is at equilibrium in another state (another set of values for temperature pressure and volume). Chemicals might react! Temperature might change! Solids might become liquids and liquids might dissolve other solids! Thermodynamics can tell you lots about energy changes in these processes and even whether the processes can actually occur or not.