# Expansion and Compression of Ideal Gases

A discussion on the expansion and compression of ideal gases, also considering the particular cases of isothermal and adiabatic processes

## Overview

**Key facts**The expansion and compression of ideal gases are polytropic processes, and therefore satisfy: where is the polytropic index. The work done during a polytropic expansion is given by: or by: where are pressures, volumes, temperatures, the mass, the universal gas constant, and the polytropic index. The work done during a polytropic compression is given by: or by: where are pressures, volumes, temperatures, the mass, the universal gas constant, and the polytropic index. The heat supplied during a polytropic expansion is given by: where is the heat capacity ratio, the polytropic index, and the work done during the expansion. The work done during an isothermal () expansion or compression can be written as: where is the mass, the universal gas constant, the temperature, the initial volume, and the final volume. The heat changed during an isothermal expansion or compression is given by: where is the work done. The work done during an adiabatic () expansion is given by: and for an adiabatic compression by: where is the change in internal energy. <br/>

**Constants**

The expansion and compression of ideal gases are regularly considered to be polytropic processes. Therefore, they satisfy the equation:
where is called the polytropic index.
However, we also know that ideal gases follow the so-called combined gas law (for a more detailed discussion also see Thermodynamics of Ideal Gases ), which states that:
Therefore, when dividing equation (1) by we will also get a constant:
or, written in a different form:
from which we obtain that, during an expansion or compression, ideal gases satisfy:
Also, we can rewrite equation (2) as:
By using the expression form of the volume from (6) in equation (1), we get that:
which leads to another equation which is satisfied during the expansion or compression of ideal gases:
Let us now consider the work done during a polytropic expansion or compression. We know that the work done by a gas which is expanding from state to state is given by:
As this is a polytropic expansion, we also have that:
or, furthermore, that:
where is a constant, and the polytropic index. By using the expression of pressure from (11) in equation (9), we get the work done by the gas as:
which, integrated, leads to:
By using identity (10) again, we can rewrite (13) as:
or, furthermore, as:
Thus, we obtain the work done by the gas during a polytropic expansion as:
However, from the ideal gas law we also have that:
where is the mass, and the universal gas constant (for additional information also see Thermodynamics of Ideal Gases ). Therefore, we have that:
and:
By using (18) and (19), the work done from equation (16) becomes:
from which we obtain the work done by the gas during a polytropic expansion also as:
In order to calculate the work done on a gas undergoing a polytropic compression from state to state , we follow a similar reasoning, but this time starting from:
Therefore, equations (16) and (21) can be rewritten in order to express the work done during a polytropic compression as:
and:
respectively. Hence, we obtain that the work done during a polytropic compression can be expressed as:
or as:
Let us now consider the heat supplied during a polytropic expansion. From the first law of thermodynamics we know that the heat added to the system equals the change in internal energy plus the work done by the system :
As the change in internal energy is given by:
where is the mass, and the heat capacity at constant volume (for a more detailed discussion also see Thermodynamics of Ideal Gases ), and also taking into account the expression of the work done during a polytropic expansion from (20), the heat supplied during a polytropic expansion becomes:
which can also be written as:
We know that the universal gas constant relates the heat capacity at constant volume to the heat capacity at constant pressure by:
(for a more detailed discussion also see Thermodynamics of Ideal Gases ). However, from the definition of the heat capacity ratio :
we also have that:
By using the expression of from (33) in equation (31), we get that:
or, furthermore, that:
from which we obtain as:
By using the expression of from (36) in equation (30), we get the heat supplied during a polytropic expansion as:
or, furthermore, as:
Equation (38) leads to:
or, furthermore, to:
Taking into account that (see equation 21), we obtain the heat supplied during a polytropic expansion as:
where is the heat capacity ratio, the polytropic index, and the work done during the expansion.

## Isothermal Expansion Or Compression

For different values of the polytropic index , the polytropic process defined by (1) will be equivalent to other particular processes. For instance, if , equation (1) will be rewritten as: However, taking into account the ideal gas law (see 17), this also means that: As and are constants, this leads to: which means that for , we are dealing with isothermal processes. As the work done during a process which changes the system from state to state is given by , and also taking into account that (see the ideal gas law, equation 17), we get the work done during an isothermal expansion or compression as: from which we obtain: Let us now consider the heat changed during an isothermal expansion or compression. The change in internal energy in this case is , as (see equation 28). Therefore, from the first law of thermodynamics (see 27), we obtain that, for an isothermal process, the heat changed equals the work done:## Adiabatic Expansion Or Compression

Another particular case of the polytropic processes is when the polytropic index equals the heat capacity ratio, i.e. when . In this case, the heat changed during the process (see equation 41) will be: Therefore, when , we are dealing with adiabatic processes. This can also be demonstrated the other way around. Consider, for example, that we are dealing with an adiabatic process. As and , we obtain from the first law of thermodynamics (see 27) written for infinitesimal changes, that: We can write the infinitesimal change in internal energy as (see equation 28): where is the mass, the heat capacity at constant volume, and the infinitesimal change in temperature. Therefore, equation (49) becomes: However, (see the ideal gas law, equation 17). Hence, we can also write equation (51) as: or, furthermore, as: By integrating equation (53) we get that: which can also be written as: or, furthermore, as: By raising to equation (56), we get that: Then, by raising equation (57) to the power, we get that: From equation (35) we can also write that: Therefore, equation (58) becomes: However, we know that all polytropic processes satisfy (see equation 5). By coupling this with equation (60) we obtain that, indeed, for adiabatic processes, . Let us now consider the work done during an adiabatic expansion or compression. As , the work done during an adiabatic expansion is given by (see equation 21): However, as (see 36), equation (61) becomes: Taking into account equation (50), we obtain the work done during an adiabatic expansion as: where is the change in internal energy. By following a similar reasoning, we obtain the work done during an adiabatic compression (given by equation 26 when ) as: where is the change in internal energy.Last Modified: 25 Sep 10 @ 08:28 Page Rendered: 2022-03-14 15:51:38