Thermal physics is the study of things like heat, temperature and the statistics of particles. Sometimes students struggle with this topic since sometimes the scientific understanding of temperature or heat is different to our intuition and day-to-day usage of those terms. So, this article will cover a definition of some of these basics concepts. This section will also not be very mathematically rigorous, since a full mathematical treatment of the subject is beyond the scope of this course.
Our first definition, heat. Heat is thermal energy in transit. This may seem simple but let's pause on this for a moment. We've used energy a lot so far in this course, but we never thought about what energy is as such. In some ways that's quite hard to answer, so we'll try to draw some analogies with kinetic energy. What is thermal energy? Well, thermal energy on a very micro-scale is actually just the movement of atoms. The physics of atoms are always very complex, and in this section we will make a very significant simplification. We will assume that atoms are just tiny hard spheres that move around. In reality, atoms are much more complex than this, but this will suffice for now.
The in transit part of the definition of heat is very important, too. If you have some object, it makes no sense to discuss how much heat it has. Heat only "exists" when thermal energy moves from one material to another. Another point here is that heat has a tendency to flow in a particular direction naturally. That is, heat moves from hotter objects to cooler objects, naturally. If we want heat to move from cooler objects into hotter objects, we need to spend energy in some way.
We've already alluded to another concept in physics - temperature. However, we used it in a very loose manner. We would like to now be more rigorous. Temperature is defined as the average kinetic energy per molecule in a substance. We apply a scale to this. Some people use the Celsius scale and some use the Fahrenheit scale. However in physics we always use the Kelvin scale. The Kelvin scale is 0K at absolute 0. This is the temperature at which an atom will no longer have any kinetic energy. Temperatures below this point are not possible. The steps in Kelvin are the same size as in the Celsius scale. In fact, the Celsius scale is simply offset from the Kelvin scale by approximately 273.15 degrees. In other words \(0^\circ C = 273.15K\)
Why do we say average kinetic energy in the definition of temperature? This is because if we have a large number of molecules, they won't all have the same kinetic energy. In fact, the kinetic energy will vary significantly between different molecules. Some will be moving very fast, and some much slower. So, we take the mean of that distribution, and that defines the temperature. We already talked of thermal energy, but now that concept should be easier to follow. Since temperature is the average kinetic energy per molecule, we can elaborate that thermal energy is the total kinetic energy of all molecules.
To really drive the point home, imagine two blocks of the same metal in thermal equilibrium. Block A weights 1kg and block B weights 10kg. They will not exchange any heat, and their temperatures at the same. Yet, block B has 10 times as much thermal energy as block A.
So, why does heat tend to flow from higher temperature materials into lower temperature materials? This is a simple application of statistics. When two materials come into contact and they are not in thermal equilibrium, it is more likely for fast moving molecules from the hot material to collide with slow moving molecules from the cold material than the other way around. This is simply a numbers game. When two molecules collide, on a micro-scale, kinetic energy will always be transferred from the faster moving molecule to the slower moving molecule. Occasionally, the colder object will transfer some heat to the hotter object, but on bulk heat will flow from the hotter object to the colder one.
Since we have thought so much about energy, let's apply the law of conservation of energy. Our first equation is
\[\Delta Q + \Delta W = \Delta U\]
\(\Delta Q\) is the heat entering an object, \(\Delta W\) is the work done on an object and \(\Delta U\) is the change in internal energy of an object.
In a basic study of thermal physics, we often deal with the ideal gas. The equation for the internal energy of the idea gas is given by
\[U = \sum_i \frac{1}{2} mv_i^2 = \frac{3}{2}NkT\]
Where \(N\) is the number of molecules, \(k\) is the Boltzmann constant, \(T\) is the temperature, \(m\) is the mass of each molecule (assumed here to be the same) and \(v_i\) is the velocity of each molecule.
There are a few thermal processes:
Adiabatic: this is a process in which \(\Delta Q = 0\). This is a process where no heat is transferred. While this isn't necessarily possible in real life, it can be approximated with processes that occur so quickly that there is insufficient time for heat transfer.
Isothermal: This is a process where there is no change in temperature, so \(\Delta U = 0\).
Isochoric: This process means there is no work done, so \(\Delta W = 0\). This would be achieved by not allowing the system to expand or contract.
Let's now take a closer look at the ideal gasses we've talked about so far. Consider an ideal gas that undergoes some thermal process. It will follow these three laws
Boyles law: When the temperature is held constant, the volume is inversely proportional to the pressure. For a thermal process, the below holds. The pressure times volume before the thermal process is the same as the pressure times volume after the thermal process.
\[p_1 V_1 = p_2 V_2\]
Charle's Law: When the pressure is held constant, the volume is proportional to the temperature. For a thermal process, we can write the following.
\[\frac{V_1}{T_1} = \frac{V_2}{T_2}\]
Gay-Lussac's Law: Given a constant volume, the pressure is proportional to the temperature. For a thermal process, we can write the following.
\[\frac{p_1}{T_1} = \frac{p_2}{T_2}\]
If we combine these three laws into one constant equation for a general thermal process on an ideal gas we find
\[\frac{p_1 V_1}{T_1} = \frac{p_2 V_2}{T_2}\]
How does the mass of each gas molecule impact the variables? Empirically, we find that
\[pV = AmT\]
Where \(A\) is some constant, that varies from gas to gas. It would be more helpful if this were put in terms of more general constants. If we use the number of moles of gas we have instead of mass, we will have
\[pV = nRT\]
Where \(n\) is the number of moles of gas and \(R\) is the molar gas constant. This is the same constant for all ideal gasses. Another form is given by
\[pV = NkT\]
Where \(N\) is the total number of molecules, and \(k\) is the Boltzmann constant. \(R\) and \(k\) are related by
\[k = \frac{R}{N_A}\]
Where \(N_A\) is Avogadro's constant
An ideal gas is defined by these equations. Specifically, if a gas obeys \(pV = nRT\) at all pressures, volumes and temperatures, then it is an ideal gas. Practically, gasses tend to approximate ideal gasses quite well as long as the temperature is far above the boiling point but the pressure is not high.