Adiabatic Processes: Definition, Equation & Examples

Thermodynamics is a branch of physics that studies processes by which heat energy can change form. Often ideal gases specifically are studied because, not only are they much simpler to understand, but many gases can be approximated as ideal.

A particular thermodynamic state is defined by state variables. These include pressure, volume and temperature. By studying the processes by which a thermodynamic system changes from one state to another, you can gain a deeper understanding of the underlying physics.

Several idealized thermodynamic processes describe how states of an ideal gas can undergo change. The adiabatic process is just one of these.

The state of an ideal gas at any one point in time can be described by the state variables pressure, volume and temperature. These three quantities are sufficient in determining the present condition of the gas and are not at all dependent on how the gas obtained its current state.

Other quantities, such as internal energy and entropy, are functions of these state variables. Again, state functions don't depend on how the system got into its particular state either. They only depend on the variables describing the state it is currently in.

Process functions, on the other hand, describe a process. Heat and work are process functions in a thermodynamic system. Heat is only exchanged during a change from one state to another, just as work can only be done as the system changes state.

An adiabatic process is a thermodynamic process that occurs with no heat transfer between the system and its environment. In other words, the state changes, work can be done on or by the system during this change, but no heat energy is added or removed.

Since no physical process can happen instantaneously and no system can truly be perfectly insulated, a perfectly adiabatic condition can never be achieved in reality. However, it can be approximated, and much can be learned by studying it.

The faster a process occurs, the closer it can be to adiabatic because the less time there will be for a transfer of heat.

The first law of thermodynamics states that the change in internal energy of a system is equal to the difference of the heat added to the system and the work done by the system. In equation form, this is:

\(\Delta E=Q-W\)

Where ​E​ is the internal energy, ​Q​ is the heat added to the system and ​W​ is the work done by the system.

Since there is no heat exchanged in an adiabatic process, then it must be the case that:

\(\Delta E=-W\)

In other words, if energy leaves the system, it is the result of the system doing work, and if energy enters the system, it results directly from work done on the system.

When a system expands adiabatically, volume increases while no heat is exchanged. This increase in volume constitutes work being done by the system on the environment. Hence the internal energy must decrease. Since internal energy is directly proportional to the temperature of the gas, this means that the temperature change will be negative (the temperature drops).

From the ideal gas law, you can get the following expression for pressure:

\(P=\frac{nRT}{V}\)

Where ​n​ is the number of moles, ​R​ is the ideal gas constant, ​T​ is temperature and ​V​ is volume.

For adiabatic expansion, temperature goes down while volume goes up. This means that pressure should also go down because, in the above expression, the numerator would decrease while the denominator would increase.

In adiabatic compression, the reverse happens. Since a decrease in volume indicates work being done on the system by the environment, this would yield a positive change in internal energy corresponding to a temperature rise (higher final temperature).

If the temperature increases while the volume decreases, then pressure also increases.

One example that illustrates an approximately adiabatic process often shown in physics courses is the operation of a fire syringe. A fire syringe consists of an insulated tube that is closed on one end and that contains a plunger on the other end. The plunger can be pushed down to compress the air in the tube.

If a small piece of cotton or other flammable material is placed in the tube at room temperature, and then the plunger is pushed down very rapidly, the state of the gas in the tube will change with minimal heat being exchanged with the outside. The increased pressure in the tube that happens upon compression causes the temperature inside the tube to rise dramatically, enough so that the small piece of cotton combusts.

A ​pressure-volume​ (P-V) diagram is a graph that depicts the change in state of a thermodynamic system. In such a diagram, volume is plotted on the ​x​-axis, and pressure is plotted on the ​y​-axis. A state is indicated by an (​x, y​) point corresponding to a particular pressure and volume. (Note: Temperature can be determined from pressure and volume using the ideal gas law).

As the state changes from one particular pressure and volume to another pressure and volume, a curve can be drawn on the diagram indicating how the state change occurred. For example, an isobaric process (in which pressure remains constant) would look like a horizontal line on a P-V diagram. Other curves can be drawn connecting the starting and ending point, and would consequently result in different amounts of work being done. This is why the shape of the path on the diagram is relevant.

An adiabatic process shows up as a curve that obeys the relationship:

\(P \propto \frac{1}{V^c}\)

Where ​c​ is the ratio of specific heats c_p/c_v (​_c_p​ is the specific heat of the gas for constant pressure, and ​c_v​ is the specific heat for constant volume). For an ideal monatomic gas, ​c​ = 1.66, and for air, which is primarily a diatomic gas, ​c_​ = 1.4

Heat engines are engines that convert heat energy into mechanical energy via a complete cycle of some sort. On a P-V diagram, a heat-engine cycle will form a closed loop, with the state of the engine ending where it started, but doing work in the process of getting there.

Many processes only work in one direction; however, reversible processes work equally well forwards and backwards without breaking the laws of physics. An adiabatic process is a type of reversible process. This makes it particularly useful in a heat engine as it means it doesn't convert any energy into an unrecoverable form.

In a heat engine, the total work done by the engine is the area contained within the loop of the cycle.

Other thermodynamic processes discussed in more detail in other articles include:

Isobaric processes, which occur at constant pressure. These will look like horizontal lines on a P-V diagram. Work done in an isobaric process is equal to the constant pressure value multiplied by the change in volume.

Isochoric process, which occur at constant volume. These look like vertical lines on a P-V diagram. Due to the fact that volume doesn't change during these processes, no work is done.

Isothermal processes occur at constant temperature. Like adiabatic processes, these are reversible. However, in order for a process to be perfectly isothermal, it must maintain a constant equilibrium, which would mean it would have to occur infinitely slowly, in contrast to the instantaneous requirement for an adiabatic process.