Thermodynamic Diagram

Thermodynamic Diagrams and the First Law

One of the main advantages of a thermodynamic diagram is that the type of process can be determined. Understanding the type of process makes it easier to calculate the change in the internal energy of the system in question. We can also use the diagram to find the changes responsible for the system's change in internal energy. For relatively simple processes, we will employ the use of the first law of thermodynamics to make these calculations.

Mathematically, we can write the first law as \[\Delta U=Q+W,\] where \(\Delta U\) represents the change in the system's internal energy, \(Q\) is the thermal energy added to the system, and \(W\) is the work done on the system. If we can determine the thermal energy and the work done, during a thermodynamic process, from a thermodynamic diagram, we could calculate the change in internal energy of that system during that process.

The work done on or by a fixed mass of gas can be determined using a PV diagram, which is simply a plot of the gas' pressure \(P\) against its volume \(V.\)

The work done \(W\) on or by a fixed mass of gas can be found from \[W=-P\Delta V,\] provided that the pressure remains constant and the volume changes. This is a special type of thermodynamic process that will discuss a little later on.

Thermodynamic Diagram Properties

Let us look at the different types of thermodynamic processes and see their associated thermodynamic diagrams. Specifically, we will examine the PV diagrams for each process.

Isothermal Processes

Isothermal processes are thermodynamic processes in which the temperature \(T\) of the system remains constant. If the system happens to be an ideal gas then the pressure \(P\) of the gas is inversely proportional to its volume \(V,\) \[P\propto \frac{1}{V},\] from Boyle's law. This relationship holds as long as the temperature and mass of the gas remain constant. A PV diagram of an isothermal process is a graph of pressure against volume and is shown in Fig. 1 below.

Fig. 1 - The PV diagram of an isothermal process shows the inverse relationship between the pressure and the volume of an ideal gas. The graph is shifted for different constant temperatures.

Note that the graph shows the inverse relationship between different pressure and volume and is shifted for different temperatures. The temperature along each line remains constant. Lines of constant temperature are called isotherms.

Isobaric Processes

Isobaric processes are thermodynamic processes in which the pressure \(P\) of the system remains constant. If the system is a gas and its volume changes during this process then we can use the equation mentioned previously, \[W=-P\Delta V,\] to find the work done \(W\) on or by the gas. Note, we can use the sign convention that \(W>0\) when work is done on the gas and \(W<0\) when the gas does external work. The PV diagram for an isobaric process is shown in Fig. 2 below.

Fig. 2 - The PV diagram for an isobaric process is represented by a horizontal line if the volume changes while the pressure remains constant.

In the PV diagram, we have a thermodynamic process \(\text{AB}\) which takes gas from state \(\text{A}\) to state \(\text{B}.\) The pressure remains constant between states but the volume changes from \(V_\mathrm{A}\) in state \(\text{A}\) to \(V_\mathrm{B}\) in state \(\text{B}.\) Since the volume is increasing, the gas is doing work externally to expand and \(W<0.\) \[\begin{align}W&=-P\Delta V\\[4 pt]&=-P\left(V_\mathrm{B}-V_\mathrm{A}\right)\\[4 pt]&=+P\left(V_\mathrm{A}-V_\mathrm{B}\right). \end{align}\]

Isovolumetric Processes

Isovolumetric (or isochoric) processes are thermodynamic processes in which the volume \(V\) of a closed system remains constant.

An example of an isovolumetric process is a gas that is contained in a container of fixed volume whilst being heated; its volume cannot change. If the volume remains constant, then \(W=0\) and \[\begin{align}W&=-P\Delta V\\[4 pt]&=-P(0)\\[4 pt]&=0. \end{align}\] From the first law, the change in internal energy \(\Delta U\) then becomes, \[\begin{align}\Delta U&=Q+0\\[4 pt]&=Q. \end{align}\] The change in the internal energy of the system is only due to an addition or removal of thermal energy from the system. If the pressure changes in this process, the PV diagram is depicted in Fig. 3 below.

Fig. 3 - The PV diagram for an isovolumetric process is represented by a vertical line if the pressure changes while the volume remains constant.

The pressure of this system changes from \(P_\mathrm{A}\) to \(P_\mathrm{B}\), as it undergoes process \(\text{AB},\) whilst the volume \(V\) remains constant.

Adiabatic Processes

Adiabatic processes are thermodynamic processes in which no thermal energy \(Q\) can enter the system. From the first law, we can see that \[\begin{align}\Delta U&=0+W\\[4 pt]&=W. \end{align}\] The change in the internal energy of the system is only due to the work done on or by the system. An example of an adiabatic process is a gas being rapidly compressed in a cylinder. The rapid compression does not allow sufficient time for the gas to gain or lose thermal energy and we say that it is being compressed adiabatically. The relationship between pressure \(P\) and volume \(V\) for an adiabatic process is as follows, \[PV^{\gamma}=k,\] where \(\gamma\) and \(k\) are both constants. From this equation, we can plot the PV diagram as in Fig. 4 below.

Fig. 4 - The PV diagram for an adiabatic process shows an inverse power relationship between pressure and volume.

The PV diagram shows that there is an inverse power relationship between the pressure and the volume of the system. The pressure decreases as the volume increases. Note that it is not immediately possible to apply the first law to this process to find the change in internal energy.

Thermodynamic Diagram Examples

We now know how to identify the different processes by their thermodynamic diagrams so we can attempt some examples to apply the first law.

We can now attempt another example relating to a different thermodynamic process.

PVT Diagrams Thermodynamics

Beyond drawing two-dimensional plots to test the relationship between two thermodynamic variables, we can also plot three variables on the same set of three-dimensional axes in order to observe the relationship between all three. This is beyond the scope of the AP physics syllabus but it is good to know that it provides more information than a PV diagram does. A simple example of this kind of plot is called a PVT diagram which plots the pressure \(P\) against the volume \(V\) and the temperature \(T.\) Fig. 7 below shows a PVT diagram for a substance that expands when freezing, such as water.