Isochoric Processes: Definition, Equation & Examples

The isochoric process is one of several idealized thermodynamic processes which describe how the states of an ideal gas can undergo change. It describes the behavior of gas in a closed container at a constant volume. In this situation, when energy is added, only the temperature of the gas changes; it does no work on its surroundings. So no motors turns, no pistons move, and no useful output happens.

What Is an Isochoric Process?

An isochoric process, (sometimes called isovolumetric or isometric process) is a thermodynamic process that occurs at a constant volume. Because the volume doesn't change, the relationship between pressure and temperature maintains a constant value.

This can be understood by starting with the ideal gas law:

PV = nRT

Where P is the absolute pressure of the gas, V is volume, n is the amount of gas, R is the ideal gas constant (8.31 J/mol K), and T is temperature.

When volume is held constant, this law can be rearranged to show that the ratio of P to T must also be a constant:

\frac {P}{T} = \text {constant}

This mathematical expression of the ratio between pressure and temperature is known as Gay-Lussac's Law, so named for the French chemist who came up with it in the early 1800s. Another outcome of this law, which is sometimes also called the pressure law, is the ability to predict temperatures and pressures for ideal gasses undergoing isochoric processes using the following equation:

\frac {P_1}{T_1} = \frac {P_2}{T_2}

Where P1 and T1 are the initial pressure and temperature of the gas, and P2 and T2 are the final values.

On a graph of pressure versus temperature, or a PV diagram, an isochoric process is represented by a vertical line.

Teflon (PTFE), the non-reactive, most slippery substance on the planet with applications across many industries from aerospace to cooking, was an accidental discovery that resulted from an isochoric process. In 1938, DuPont chemist Roy Plunkett had set up a bunch of small cylinders to store tetrafluoroethylene gas, for use in refrigeration technologies, which he then cooled to an extremely low temperature.

When Plunkett went to open one later, no gas came out, though the mass of the cylinder hadn't changed. He chopped open the tube to investigate and saw a white powder coating the inside, which later proved to have immensely useful commercial properties.

According to Gay-Lussac's Law, when the temperature rapidly decreased, so did the pressure to initiate a phase change in the gas.

Isochoric Processes and The First Law of Thermodynamics

The first law of thermodynamics states that the change in the internal energy of a system is equal to the heat added to the system minus the work done by the system. (In other words, energy input minus energy output.)

The work done by an ideal gas is defined as its pressure times its change in volume, or PΔV (or PdV). Because the volume change ΔV, is zero in an isochoric process, however, no work is done by the gas.

Hence, the change in internal energy of the gas is simply equal to the amount of heat added.

An example of a nearly isochoric process is a pressure cooker. When sealed closed, the volume inside cannot change, so when heat is added both pressure and temperature increase rapidly. In actuality, pressure cookers do expand slightly, and some gas is released from a valve on top.

Isochoric Processes in Heat Engines

Heat engines are devices that harness the the transfer of heat to do some kind of work. They use a cyclical system to convert heat energy added to them into mechanical energy, or motion. Examples include steam turbines and automotive engines.

Isochoric processes are used in many common heat engines. The Otto Cycle, for example, is a thermodynamic cycle in car engines that describes the process of heat transfer during ignition, the power stroke moving engine pistons to make the car go, the release of heat, and the compression stroke returning pistons to their starting positions.

In the Otto Cycle, the first and third steps, the addition and release of heat, are considered isochoric processes. The cycle assumes the heat changes occur instantaneously, with no change in volume of the gas. Thus, work is only done on the vehicle during the power and compression stroke phases.

The work done by a heat engine using the Otto cycle is represented by the area under the curve in the diagram. This is zero where the isochoric processes of heat addition and release are occurring (the vertical lines).

Isochoric processes like these are generally irreversible processes. Once heat is added, the only way to return the system to its original state is to remove heat somehow by doing work.

Other Thermodynamic Processes

Isochoric processes are but one of several idealized thermodynamic processes that describe the behavior of gasses useful to scientists and engineers.

Some of the others discussed in more detail elsewhere on the site include:

Isobaric process: This occurs at a constant pressure and is common in many real life examples, including boiling water on a stove, lighting a match or in air-breathing jet turbines. This is because, for the most part, the pressure of the Earth's atmosphere is not changing a lot in a local area, such as the kitchen in which someone is making pasta. Assuming the ideal gas law applies, temperature divided by volume is a constant value for an isobaric process.

Isothermal process: This occurs at a constant temperature. For example, during a phase change such as water boiling off the top of a pot, the temperature is steady. Refrigerators also use isothermal processes and an industrial application is the Carnot Engine. Such a process is slow because the heat added must be equal to the heat lost as work in order to keep the overall temperature constant. Assuming the ideal gas law applies, pressure times volume is a constant value for an isothermal process.

Adiabatic process: There is no heat or material exchange with the surroundings as a gas or fluid changes volume. Instead, the only output in an adiabatic process is work. There are two cases in which an adiabatic process might occur. Either, the process occurs too quickly for heat to transfer in or out of the whole system, such as during the compression stroke of a gas engine, or it happens in a container that is so well insulated heat cannot cross the barrier at all.

Like the other thermodynamics processes explained here, no process is truly adiabatic, but approximating against this ideal is useful in physics and engineering. For instance, a common characterization for compressors, turbines and other thermodynamic machines is adiabatic efficiency: The ratio of the actual work the machine outputs to how much work it would output if it underwent a true adiabatic process.

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