Thermodynamic Diagram

Imagine the ecstasy and elation early humans must have felt when they discovered how to create and control fire. The ability able to heat cold spaces and cook food was probably the first significant discovery of humankind. Rock art, ancient papyrus scrolls, and, more recently, the vast expanse of the internet all contain information about this discovery and its importance. Since language was not a method of communication for our early ancestors, they had to spread the word about their amazing discovery. The next best method of communication would be a picture or diagram of the fire-making instructions. Fast-forward to the modern-day world and we still find that charts, pictures, and diagrams are probably the most effective ways of vital communication. In this article, we will learn about how thermodynamic diagrams can be used to identify various properties of thermodynamic systems.

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What is the relationship between pressure \(P\) and volume \(V\) for an adiabatic process?

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A thermodynamic diagram is a diagram that illustrates only one thermodynamic variable during a thermodynamic process.

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What is the work done to compress a gas that remains at a constant pressure of \(3.0\,\mathrm{kPa}\) and undergoes a change in volume of \(1.5\,\mathrm{m^3}?\)

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What is the equation for the work done \(W\) on or by a fixed mass of gas at constant pressure \(P\) with a change in volume \(\Delta V\)?

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\(800\,\mathrm{J}\) of thermal energy enters a fixed mass of an ideal gas. What is the change in internal energy of the gas if it undergoes a change in pressure of \(2.0\,\mathrm{kPa}\) at a constant volume of \(5.0\times 10^{-4}\,\mathrm{m^3}?\)

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What does the internal energy \(\Delta U\) simplify to, in the case of an isovolumetric process?

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Which three thermodynamic variables are plotted against each other in a PVT diagram?

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What does the internal energy \(\Delta U\) simplify to, in the case of an adiabatic process?

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No work can be done on or by an ideal gas that undergoes an adiabatic process.

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What is the relationship between pressure \(P\) and volume \(V\) for an adiabatic process?

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  • Immunology
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A thermodynamic diagram is a diagram that illustrates only one thermodynamic variable during a thermodynamic process.

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What is the work done to compress a gas that remains at a constant pressure of \(3.0\,\mathrm{kPa}\) and undergoes a change in volume of \(1.5\,\mathrm{m^3}?\)

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What is the equation for the work done \(W\) on or by a fixed mass of gas at constant pressure \(P\) with a change in volume \(\Delta V\)?

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  • Immunology
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  • Mo

\(800\,\mathrm{J}\) of thermal energy enters a fixed mass of an ideal gas. What is the change in internal energy of the gas if it undergoes a change in pressure of \(2.0\,\mathrm{kPa}\) at a constant volume of \(5.0\times 10^{-4}\,\mathrm{m^3}?\)

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What does the internal energy \(\Delta U\) simplify to, in the case of an isovolumetric process?

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Which three thermodynamic variables are plotted against each other in a PVT diagram?

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What does the internal energy \(\Delta U\) simplify to, in the case of an adiabatic process?

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No work can be done on or by an ideal gas that undergoes an adiabatic process.

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    Thermodynamic Diagram Definition

    There are different types of thermodynamic processes, each of which takes a thermodynamic system between states. It is almost always the case that a change in one of the thermodynamic variables (pressure, entropy, temperature, etc.) will lead to a change in another. It would be beneficial to be able to visualize the relationship between the two variables and to do this we employ thermodynamic diagrams.

    A thermodynamic diagram is a diagram that illustrates the relationship between two or more thermodynamic variables during a thermodynamic process.

    The diagram usually takes the form of a graph in which two variables of state are plotted against each other. For example, temperature and entropy could be plotted on the same set of axes to test their relationship during a thermodynamic process. The more common comparison is that between the pressure and volume of an ideal gas, which is the relationship that we will focus on, in this article.

    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.

    The first law of thermodynamics states that the increase in internal energy of a system is equal to the sum of the thermal energy added to the system and the work done on the system.

    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.

    Thermodynamic diagrams PV diagram of isothermal process StudySmarterFig. 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.

    Thermodynamic diagrams PV diagram of an isobaric process StudySmarterFig. 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.

    In thermodynamics, a closed system is one in which no mass is allowed to enter or leave the system.

    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.

    Thermodynamic diagrams PV diagram for isovolumetric process StudySmarterFig. 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.

    Thermodynamic diagrams PV diagram for adiabatic process StudySmarterFig. 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.

    The constant \(\gamma\) is known as the ratio of specific heats of the gas undergoing the adiabatic process, \[\gamma=\frac{C_P}{C_V},\] where \(C_P\) is the specific heat of the gas at constant pressure and \(C_V\) is the specific heat of the gas at constant volume. These values depend on the gas in the system, e.g. \(\gamma_\text{air}=1.4.\)

    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.

    Question: A fixed mass of an ideal gas undergoes an isobaric expansion process \(\text{AB}\), shown in Fig. 5 below, at a constant pressure of \(4.0\times 10^{5}\,\mathrm{Pa}.\) During this process, its volume increases from \(2.0\times 10^{-4}\,\mathrm{m^3}\) to \(8.0\times 10^{-4}\,\mathrm{m^3}.\) Find the change in the internal energy of the gas during this process if no thermal energy enters or leaves the gas.

    Thermodynamic diagrams Isobaric PV diagram example StudySmarterFig. 5 - The PV diagram for the isobaric process involving the ideal gas in the example.

    Answer: Since the pressure is constant and the volume is changing, we can calculate the work done. The gas is expanding and so it is doing work against its surroundings, \[\begin{align} W&=-P\Delta V\\[4 pt]&=-\left(4.0\times 10^{5}\,\mathrm{Pa}\right)\left(8.0\times 10^{-4}\,\mathrm{m^3} - 2.0\times 10^{-4}\,\mathrm{m^3}\right)\\[4 pt]&=-240\,\mathrm{J}. \end{align}\] This means that the gas does \(240\,\mathrm{J}\) of work during expansion. Since there is no thermal energy transfer, \(Q=0,\) and we can use the first law to find the change in internal energy of the gas. \[\begin{align} \Delta U&=Q+W\\&=0+(-240\,\mathrm{J})\\&=-240\,\mathrm{J}. \end{align}\] The internal energy change is negative which means that the internal energy of the gas decreases by \(240\,\mathrm{J}.\) This makes sense since the gas must lose energy by doing work.

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

    Question: A fixed mass of an ideal gas undergoes an isovolumetric process \(\text{AB}\) at a constant volume of \(3.0\times 10^{-4}\,\mathrm{m^3}.\) During this process, the pressure of the gas increases from \(3.0\times 10^{5}\,\mathrm{Pa}\) to \(8.0\times 10^{5}\,\mathrm{Pa}.\) Thermal energy of \(500\,\mathrm{J}\) enters the gas during \(\text{AB}.\) Calculate the change in internal energy of the gas during this process. The PV diagram for this process is shown in Fig. 6 below.

    Thermodynamic diagrams Isovolumetric PV diagram example StudySmarterFig. 6 - The PV diagram for the isovolumetric process involving the ideal gas in the example.

    Answer: We can attempt to find the work done for the ideal gas as follows, \[\begin{align} W&=-P\Delta V\\[4 pt]&=-P(0)\\[4 pt]&=0\,\mathrm{J.} \end{align}\] The gas does no work nor is any work done on it since its volume is remaining constant. \(500 \,\mathrm{J}\) of thermal energy enters the gas and so we have \(Q=500\,\mathrm{J}.\) Applying the first law of thermodynamics, \[\begin{align} \Delta U&=Q+W\\[4 pt]&=500\,\mathrm{J}+0\\[4 pt]&=500\,\mathrm{J}. \end{align}\] The internal energy change is positive which means that the internal energy of the gas increases by \(500\,\mathrm{J}.\)

    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.

    Thermodynamic Diagram - Key takeaways

    • A thermodynamic diagram is a diagram that illustrates the relationship between two or more thermodynamic variables during a thermodynamic process.
    • The first law of thermodynamics states that the increase in internal energy \(\Delta U\) of a system is equal to the sum of the thermal energy \(Q\) added to the system and the work done \(W\) on the system, \[\Delta U =Q+W.\]
    • The work done \(W\) on or by a fixed mass of gas can be found from \[W=-P\Delta V,\] provided that the pressure \(P\) remains constant and the volume changes by \(\Delta V\).
    • Isothermal processes are thermodynamic processes in which the temperature \(T\) of the system remains constant.
    • Isobaric processes are thermodynamic processes in which the pressure \(P\) of the system remains constant.
    • Isovolumetric processes are thermodynamic processes in which the volume \(V\) of a closed system remains constant.
    • Adiabatic processes are thermodynamic processes in which no thermal energy \(Q\) can enter the system.
    • A PVT diagram plots the pressure \(P\) against the volume \(V\) and the temperature \(T.\)
    Frequently Asked Questions about Thermodynamic Diagram

    What is property diagram in thermodynamics?

    Property diagrams are diagrams that relate the phase of a substance with its property.

    What are different types of thermodynamic diagrams?

    There are different types of thermodynamic diagrams. The main ones are PV diagrams and TS diagrams.

    How do you draw a thermodynamic diagram?

    Thermodynamic diagrams are drawn by plotting the thermodynamic variables across the axes.

    What is the basic concept of thermodynamic diagram?

    Thermodynamic diagrams are based on the properties of thermodynamics, which is the relation between heat, temperature, work, and energy.

    What is TS diagram in thermodynamics? 

    TS diagrams are thermodynamic diagrams that visualize the change in the temperature with the entropy in a thermodynamic process.

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    What is the relationship between pressure \(P\) and volume \(V\) for an adiabatic process?

    A thermodynamic diagram is a diagram that illustrates only one thermodynamic variable during a thermodynamic process.

    What is the work done to compress a gas that remains at a constant pressure of \(3.0\,\mathrm{kPa}\) and undergoes a change in volume of \(1.5\,\mathrm{m^3}?\)

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