Expansion and Compression of Ideal Gases

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.