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 $n$ 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 (13) by $\displaystyle \frac{PV}{T}$ 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 (14) as:

By using the expression form of the volume from (18) in equation (13), 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 $1$ to state $2$ is given by:

As this is a polytropic expansion, we also have that:

or furthermore, that:

where $C$ is a constant and $n$ the polytropic index. By using the expression of pressure from (23) in equation (21), we get the work done by the gas as:

which, integrated, leads to:

By using identity (22) again, we can rewrite (25) 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 $m$ is the number of moles, and $\tilde{R}$ the universal gas constant (for additional information also see Thermodynamics of Ideal Gases ). Therefore, we have that:

and:

By using (30) and (31), the work done from equation (28) 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 $1$ to state $2$, we follow a similar reasoning, but this time starting from:

Therefore, equations (28) and (33) 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 $Q$ equals the change in internal energy $\Delta U$ plus the work done by the system $W$:

As the change in internal energy is given by:

where $m$ is the number of moles, and $C_V$ the molar heat capacity at constant volume (for a more detailed discussion also see Thermodynamics of Ideal Gases ). Also taking into account the expression of the work done during a polytropic expansion from (32), the heat supplied during a polytropic expansion becomes:

which can also be written as:

We know that the universal gas constant $\tilde{R}$ relates the heat capacity at constant volume $C_V$ to the heat capacity at constant pressure $C_P$ by:

(for a more detailed discussion also see Thermodynamics of Ideal Gases ). However, from the definition of the heat capacity ratio $\gamma$:

we also have that:

By using the expression of $C_P$ from (45) in equation (43), we get that:

or furthermore, that:

from which we obtain $C_V$ as:

By using the expression of $C_V$ from (48) in equation (42), we get the heat supplied during a polytropic expansion as:

or furthermore, as:

Equation (50) leads to:

or, furthermore, to:

Taking into account that $\displaystyle \frac{m \tilde{R} (T_2 - T_1)}{1-n} = W$ (see equation 33), we obtain the heat supplied during a polytropic expansion as:

where $\gamma$ is the heat capacity ratio, $n$ the polytropic index, and $W$ the work done during the expansion.

Isothermal Expansion or Compression

For different values of the polytropic index $n$, the polytropic process defined by (13) will be equivalent to other particular processes. For instance, if $n=1$, equation (13) will be rewritten as:

However, taking into account the ideal gas law (see 29), this also means that:

As $m$ and $\tilde{R}$ are constants, this leads to:

which means that for $n=1$, we are dealing with isothermal processes.

The work done during a process which changes the system from state $1$ to state $2$ is given by $W=\int_1^2 PdV$. In addition, taking into account that $\displaystyle P=\frac{m \tilde{R} T}{V}$ (see the ideal gas law from equation 29), we get the work done during an isothermal expansion or compression as:

from which we obtain:

Let us now consider the heat exchanged during an isothermal expansion or compression. The change in internal energy in this case is $\Delta U = 0$, as $T_1=T_2$ (see equation 40). Therefore, from the first law of thermodynamics (see 39), we obtain that for an isothermal process, the heat exchanged 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 $n=\gamma$. In this case, the heat exchanged during the process (see equation 53) will be:

Therefore, when $n=\gamma$, 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 $\delta Q = 0$ and $\delta W = PdV$, we obtain from the first law of thermodynamics (see 39) written for infinitesimal changes, that:

We can write the infinitesimal change in internal energy as (see equation 40):

where $m$ is the number of moles, $C_V$ the molar heat capacity at constant volume, and $dT$ the infinitesimal change in temperature. Therefore, equation (61) becomes:

However, $\displaystyle P=\frac{m \tilde{R} T}{V}$ (see the ideal gas law from equation 29). Hence, we can also write equation (63) as:

or furthermore, as:

By integrating equation (65) we get that:

which can also be written as:

or furthermore, as:

By raising $e$ to equation (68), we get that:

Then, by raising equation (69) to the $\displaystyle \frac{1}{C_V}$ power, we get that:

From equation (47) we can also write that:

Therefore, equation (70) becomes:

However, we know that all polytropic processes satisfy $V^{n-1} \cdot T = const.$ (see equation 17). By coupling this with equation (72) we obtain that, indeed for adiabatic processes, $n=\gamma$.

Let us now consider the work done during an adiabatic expansion or compression. As $n=\gamma$, the work done during an adiabatic expansion is given by (see equation 33):

However, as $\displaystyle \frac{\tilde{R}}{1-\gamma} = - C_V$ (see 48), equation (73) becomes:

Taking into account equation (62), we obtain the work done during an adiabatic expansion as:

where $\Delta U$ is the change in internal energy.

By following a similar reasoning, we obtain the work done during an adiabatic compression (given by equation 38 when $n=\gamma$) as:

where $\Delta U$ is the change in internal energy.