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# 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
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## Overview

Key facts

The expansion and compression of ideal gases are polytropic processes, and therefore satisfy:

$PV^n&space;=&space;const.$

where $\inline&space;n$ is the polytropic index.

The work done during a polytropic expansion is given by:

$W&space;=&space;\frac{P_2&space;V_2&space;-&space;P_1&space;V_1}{1-n}$

or by:

$W&space;=&space;\frac{m&space;\tilde{R}&space;(T_2&space;-&space;T_1)}{1-n}$

where $\inline&space;P$ are pressures, $\inline&space;V$ volumes, $\inline&space;T$ temperatures, $\inline&space;m$ the mass, $\inline&space;\tilde{R}$ the universal gas constant, and $\inline&space;n$ the polytropic index.

The work done during a polytropic compression is given by:

$W&space;=&space;\frac{P_1&space;V_1&space;-&space;P_2&space;V_2}{1-n}$

or by:

$W&space;=&space;\frac{m&space;\tilde{R}&space;(T_1&space;-&space;T_2)}{1-n}$

where $\inline&space;P$ are pressures, $\inline&space;V$ volumes, $\inline&space;T$ temperatures, $\inline&space;m$ the mass, $\inline&space;\tilde{R}$ the universal gas constant, and $\inline&space;n$ the polytropic index.

The heat supplied during a polytropic expansion is given by:

$Q&space;=&space;W&space;\frac{(\gamma&space;-&space;n)}{\gamma&space;-&space;1}$

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

The work done during an isothermal ($\inline&space;n=1$) expansion or compression can be written as:

$W&space;=&space;m&space;\tilde{R}&space;T&space;ln&space;\frac{V_2}{V_1}$

where $\inline&space;m$ is the mass, $\inline&space;\tilde{R}$ the universal gas constant, $\inline&space;T$ the temperature, $\inline&space;V_1$ the initial volume, and $\inline&space;V_2$ the final volume.

The heat changed during an isothermal expansion or compression is given by:

$Q&space;=&space;W$

where $\inline&space;W$ is the work done.

The work done during an adiabatic ($\inline&space;n&space;=&space;\gamma$) expansion is given by:

$W&space;=&space;-&space;\Delta&space;U$

and for an adiabatic compression by:

$W&space;=&space;\Delta&space;U$

where $\inline&space;\Delta&space;U$ is the change in internal energy.

<br/>

Constants

$\tilde{R}&space;=&space;1543.349&space;\;&space;\frac{ft-lb}{lb-mol&space;^\circ&space;F}$

$\tilde{R}&space;=&space;8.31447&space;\;&space;\frac{J}{mol&space;K}$

The expansion and compression of ideal gases are regularly considered to be polytropic processes. Therefore, they satisfy the equation:

$PV^n&space;=&space;const.$

where $\inline&space;n$ 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:

$\frac{PV}{T}&space;=&space;const.$

Therefore, when dividing equation (2) by $\inline&space;\displaystyle&space;\frac{PV}{T}$ we will also get a constant:

$PV^n&space;\div&space;\frac{PV}{T}&space;=&space;const.$

or, written in a different form:

$PV^n&space;\cdot&space;\frac{T}{PV}&space;=&space;const.$

from which we obtain that, during an expansion or compression, ideal gases satisfy:

$TV^{n-1}&space;=&space;const.$

Also, we can rewrite equation (3) as:

$V&space;=&space;const.&space;\cdot&space;\frac{T}{P}$

By using the expression form of the volume from (7) in equation (2), we get that:

$\frac{PT^n}{P^n}&space;=&space;const.$

which leads to another equation which is satisfied during the expansion or compression of ideal gases:

$\frac{T}{P^\frac{n-1}{n}}&space;=&space;const.$

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 $\inline&space;1$ to state $\inline&space;2$ is given by:

$W&space;=&space;\int_1^2&space;PdV$

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

$PV^n&space;=&space;C$

or, furthermore, that:

$P&space;=&space;\frac{C}{V^n}$

where $\inline&space;C$ is a constant, and $\inline&space;n$ the polytropic index. By using the expression of pressure from (12) in equation (10), we get the work done by the gas as:

$W&space;=&space;C&space;\int_1^2&space;V^{-n}&space;dV$

which, integrated, leads to:

$W&space;=&space;C&space;\left(&space;\frac{V^{-n+1}}{-n+1}&space;\right)&space;\bigg|_1^2$

By using identity (11) again, we can rewrite (14) as:

$W&space;=&space;\left(&space;PV^n&space;\frac{V^{1-n}}{1-n}&space;\right)&space;\bigg|_1^2$

or, furthermore, as:

$W&space;=&space;\left(&space;\frac{PV}{1-n}&space;\right)&space;\bigg|_1^2$

Thus, we obtain the work done by the gas during a polytropic expansion as:

$W&space;=&space;\frac{P_2&space;V_2&space;-&space;P_1&space;V_1}{1-n}$

However, from the ideal gas law we also have that:

$PV&space;=&space;m&space;\tilde{R}&space;T$

where $\inline&space;m$ is the mass, and $\inline&space;\tilde{R}$ the universal gas constant (for additional information also see Thermodynamics of Ideal Gases ). Therefore, we have that:

$P_1&space;V_1&space;=&space;m&space;\tilde{R}&space;T_1$

and:

$P_2&space;V_1&space;=&space;m&space;\tilde{R}&space;T_2$

By using (19) and (20), the work done from equation (17) becomes:

$W&space;=&space;\frac{m&space;\tilde{R}&space;T_2&space;-&space;m&space;\tilde{R}&space;T_1}{1-n}$

from which we obtain the work done by the gas during a polytropic expansion also as:

$W&space;=&space;\frac{m&space;\tilde{R}&space;(T_2&space;-&space;T_1)}{1-n}$

In order to calculate the work done on a gas undergoing a polytropic compression from state $\inline&space;1$ to state $\inline&space;2$, we follow a similar reasoning, but this time starting from:

$W&space;=&space;-&space;\int_1^2&space;PdV$

Therefore, equations (17) and (22) can be rewritten in order to express the work done during a polytropic compression as:

$W&space;=&space;-&space;\frac{(P_2&space;V_2&space;-&space;P_1&space;V_1)}{1-n}$

and:

$W&space;=&space;-&space;\frac{m&space;\tilde{R}&space;(T_2&space;-&space;T_1)}{1-n}$

respectively. Hence, we obtain that the work done during a polytropic compression can be expressed as:

$W&space;=&space;\frac{P_1&space;V_1&space;-&space;P_2&space;V_2}{1-n}$

or as:

$W&space;=&space;\frac{m&space;\tilde{R}&space;(T_1&space;-&space;T_2)}{1-n}$

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 $\inline&space;Q$ equals the change in internal energy $\inline&space;\Delta&space;U$ plus the work done by the system $\inline&space;W$:

$Q&space;=&space;\Delta&space;U&space;+&space;W$

As the change in internal energy is given by:

$\Delta&space;U&space;=&space;m&space;C_V&space;(T_2&space;-&space;T_1)$

where $\inline&space;m$ is the mass, and $\inline&space;C_V$ 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 (21), the heat supplied during a polytropic expansion becomes:

$Q&space;=&space;m&space;C_V&space;(T_2&space;-&space;T_1)&space;+&space;\frac{m&space;\tilde{R}&space;(T_2&space;-&space;T_1)}{1-n}$

which can also be written as:

$Q&space;=&space;(T_2&space;-&space;T_1)&space;\left(&space;m&space;C_V&space;+&space;\frac{m&space;\tilde{R}}{1-n}&space;\right)$

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

$C_P&space;-&space;C_V&space;=&space;\tilde{R}$

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

$\gamma&space;=&space;\frac{C_P}{C_V}$

we also have that:

$C_P&space;=&space;\gamma&space;C_V$

By using the expression of $\inline&space;C_P$ from (34) in equation (32), we get that:

$\gamma&space;C_V&space;-&space;C_V&space;=&space;\tilde{R}$

or, furthermore, that:

$C_V&space;(\gamma&space;-&space;1)&space;=&space;\tilde{R}$

from which we obtain $\inline&space;C_V$ as:

$C_V&space;=&space;\frac{\tilde{R}}{\gamma&space;-&space;1}$

By using the expression of $\inline&space;C_V$ from (37) in equation (31), we get the heat supplied during a polytropic expansion as:

$Q&space;=&space;(T_2&space;-&space;T_1)&space;\left(&space;\frac{m&space;\tilde{R}}{\gamma&space;-&space;1}&space;+&space;\frac{m&space;\tilde{R}}{1-n}&space;\right)$

or, furthermore, as:

$Q&space;=&space;m&space;\tilde{R}&space;(T_2&space;-&space;T_1)&space;\left(&space;\frac{1}{\gamma&space;-&space;1}&space;+&space;\frac{1}{1-n}&space;\right)$

Equation (39) leads to:

$Q&space;=&space;m&space;\tilde{R}&space;(T_2&space;-&space;T_1)&space;\left[&space;\frac{1-n+\gamma&space;-&space;1}{(\gamma&space;-1)(1-n)}&space;\right]$

or, furthermore, to:

$Q&space;=&space;m&space;\tilde{R}&space;(T_2&space;-&space;T_1)&space;\frac{(\gamma&space;-&space;n)}{(\gamma&space;-&space;1)(1-n)}$

Taking into account that $\inline&space;\displaystyle&space;\frac{m&space;\tilde{R}&space;(T_2&space;-&space;T_1)}{1-n}&space;=&space;W$ (see equation 22), we obtain the heat supplied during a polytropic expansion as:

$Q&space;=&space;W&space;\frac{(\gamma&space;-&space;n)}{\gamma&space;-&space;1}$

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

## Isothermal Expansion Or Compression

For different values of the polytropic index $\inline&space;n$, the polytropic process defined by (2) will be equivalent to other particular processes. For instance, if $\inline&space;n=1$, equation (2) will be rewritten as:

$PV&space;=&space;const.$

However, taking into account the ideal gas law (see 18), this also means that:

$m&space;\tilde{R}&space;T&space;=&space;const.$

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

$T&space;=&space;const.$

which means that for $\inline&space;n=1$, we are dealing with isothermal processes.

As the work done during a process which changes the system from state $\inline&space;1$ to state $\inline&space;2$ is given by $\inline&space;W=\int_1^2&space;PdV$, and also taking into account that $\inline&space;\displaystyle&space;P=\frac{m&space;\tilde{R}&space;T}{V}$ (see the ideal gas law, equation 18), we get the work done during an isothermal expansion or compression as:

$W&space;=&space;\int_1^2&space;\frac{m&space;\tilde{R}&space;T}{V}&space;dV$

from which we obtain:

$W&space;=&space;m&space;\tilde{R}&space;T&space;ln&space;\frac{V_2}{V_1}$

Let us now consider the heat changed during an isothermal expansion or compression. The change in internal energy in this case is $\inline&space;\Delta&space;U&space;=&space;0$, as $\inline&space;T_1=T_2$ (see equation 29). Therefore, from the first law of thermodynamics (see 28), we obtain that, for an isothermal process, the heat changed equals the work done:

$Q&space;=&space;W$

## Adiabatic Expansion Or Compression

Another particular case of the polytropic processes is when the polytropic index equals the heat capacity ratio, i.e. when $\inline&space;n=\gamma$. In this case, the heat changed during the process (see equation 42) will be:

$Q&space;=&space;0$

Therefore, when $\inline&space;n=\gamma$, 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 $\inline&space;\delta&space;Q&space;=&space;0$ and $\inline&space;\delta&space;W&space;=&space;PdV$, we obtain from the first law of thermodynamics (see 28) written for infinitesimal changes, that:

$PdV&space;=&space;-dU$

We can write the infinitesimal change in internal energy as (see equation 29):

$dU&space;=&space;m&space;C_V&space;dT$

where $\inline&space;m$ is the mass, $\inline&space;C_V$ the heat capacity at constant volume, and $\inline&space;dT$ the infinitesimal change in temperature. Therefore, equation (50) becomes:

$PdV&space;=&space;-&space;m&space;C_V&space;dT$

However, $\inline&space;\displaystyle&space;P=\frac{m&space;\tilde{R}&space;T}{V}$ (see the ideal gas law, equation 18). Hence, we can also write equation (52) as:

$\frac{m&space;\tilde{R}&space;T}{V}&space;dV&space;=&space;-m&space;C_V&space;dT$

or, furthermore, as:

$\tilde{R}&space;\frac{dV}{V}&space;=&space;-&space;C_V&space;\frac{dT}{T}$

By integrating equation (54) we get that:

$R&space;ln&space;V&space;+&space;C_V&space;ln&space;T&space;=&space;const.$

which can also be written as:

$ln&space;V^R&space;+&space;ln&space;T^{C_V}&space;=&space;const.$

or, furthermore, as:

$ln&space;(V^R&space;\cdot&space;T^{C_V})&space;=&space;const.$

By raising $\inline&space;e$ to equation (57), we get that:

$V^R&space;\cdot&space;T^{C_V}&space;=&space;const.$

Then, by raising equation (58) to the $\inline&space;\displaystyle&space;\frac{1}{C_V}$ power, we get that:

$V^\frac{\tilde{R}}{C_V}&space;\cdot&space;T&space;=&space;const.$

From equation (36) we can also write that:

$\frac{\tilde{R}}{C_V}&space;=&space;\gamma&space;-&space;1$

Therefore, equation (59) becomes:

$V^{\gamma&space;-&space;1}&space;\cdot&space;T&space;=&space;const.$

However, we know that all polytropic processes satisfy $\inline&space;V^{n-1}&space;\cdot&space;T&space;=&space;const.$ (see equation 6). By coupling this with equation (61) we obtain that, indeed, for adiabatic processes, $\inline&space;n=\gamma$.

Let us now consider the work done during an adiabatic expansion or compression. As $\inline&space;n=\gamma$, the work done during an adiabatic expansion is given by (see equation 22):

$W&space;=&space;\frac{m&space;\tilde{R}&space;(T_2&space;-&space;T_1)}{1-\gamma}$

However, as $\inline&space;\displaystyle&space;\frac{\tilde{R}}{1-\gamma}&space;=&space;-&space;C_V$ (see 37), equation (62) becomes:

$W&space;=&space;-m&space;C_V&space;(T_2&space;-&space;T_1)$

Taking into account equation (51), we obtain the work done during an adiabatic expansion as:

$W&space;=&space;-\Delta&space;U$

where $\inline&space;\Delta&space;U$ is the change in internal energy.

By following a similar reasoning, we obtain the work done during an adiabatic compression (given by equation 27 when $\inline&space;n=\gamma$) as:

$W&space;=&space;\Delta&space;U$

where $\inline&space;\Delta&space;U$ is the change in internal energy.