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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

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 (1) 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 (2) as:

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

By using the expression form of the volume from (6) in equation (1), 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 (11) in equation (9), we get the work done by the gas as:

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

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

By using identity (10) again, we can rewrite (13) 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 (18) and (19), the work done from equation (16) 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 (16) and (21) 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 (20), 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 (33) in equation (31), 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 (36) in equation (30), 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:

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

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

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 21), 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 (1) will be equivalent to other particular processes. For instance, if $\inline&space;n=1$, equation (1) will be rewritten as:

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

However, taking into account the ideal gas law (see 17), 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 17), 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 28). Therefore, from the first law of thermodynamics (see 27), we obtain that, for an isothermal process, the heat changed equals the work done:

$Q&space;=&space;W$

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 41) 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 27) written for infinitesimal changes, that:

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

We can write the infinitesimal change in internal energy as (see equation 28):

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

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 (49) 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 17). Hence, we can also write equation (51) 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 (53) 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 (56), we get that:

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

Then, by raising equation (57) 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 (35) we can also write that:

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

Therefore, equation (58) 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 5). By coupling this with equation (60) 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 21):

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

However, as $\inline&space;\displaystyle&space;\frac{\tilde{R}}{1-\gamma}&space;=&space;-&space;C_V$ (see 36), equation (61) becomes:

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

Taking into account equation (50), 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 26 when $\inline&space;n=\gamma$) as:

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

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