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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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Contents

Overview
Isothermal Expansion Or Compression
Adiabatic Expansion Or Compression
Page Comments

Overview

Key facts

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

$PV^n = const.$

where $\inline n$ is the polytropic index.

The work done during a polytropic expansion is given by:

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

or by:

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

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

The work done during a polytropic compression is given by:

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

or by:

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

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

The heat supplied during a polytropic expansion is given by:

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

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

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

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

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

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

$Q = W$

where $\inline W$ is the work done.

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

$W = - \Delta U$

and for an adiabatic compression by:

$W = \Delta U$

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

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Constants

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

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

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

$PV^n = const.$

(2)

where $\inline n$ 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:

$\frac{PV}{T} = const.$

(3)

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

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

(4)

or, written in a different form:

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

(5)

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

$TV^{n-1} = const.$

(6)

Also, we can rewrite equation (3) as:

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

(7)

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

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

(8)

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

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

(9)

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 $\inline 1$ to state $\inline 2$ is given by:

$W = \int_1^2 PdV$

(10)

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

$PV^n = C$

(11)

or, furthermore, that:

$P = \frac{C}{V^n}$

(12)

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

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

(13)

which, integrated, leads to:

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

(14)

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

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

(15)

or, furthermore, as:

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

(16)

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

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

(17)

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

$PV = m \tilde{R} T$

(18)

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

$P_1 V_1 = m \tilde{R} T_1$

(19)

and:

$P_2 V_1 = m \tilde{R} T_2$

(20)

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

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

(21)

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

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

(22)

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

$W = - \int_1^2 PdV$

(23)

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

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

(24)

and:

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

(25)

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

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

(26)

or as:

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

(27)

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

$Q = \Delta U + W$

(28)

As the change in internal energy is given by:

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

(29)

where $\inline m$ is the mass, and $\inline C_V$ 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:

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

(30)

which can also be written as:

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

(31)

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

$C_P - C_V = \tilde{R}$

(32)

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

$\gamma = \frac{C_P}{C_V}$

(33)

we also have that:

$C_P = \gamma C_V$

(34)

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

$\gamma C_V - C_V = \tilde{R}$

(35)

or, furthermore, that:

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

(36)

from which we obtain $\inline C_V$ as:

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

(37)

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

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

(38)

or, furthermore, as:

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

(39)

Equation (39) leads to:

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

(40)

or, furthermore, to:

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

(41)

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

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

(42)

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

Isothermal Expansion Or Compression

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

$PV = const.$

(43)

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

$m \tilde{R} T = const.$

(44)

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

$T = const.$

(45)

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

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

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

(46)

from which we obtain:

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

(47)

Let us now consider the heat changed during an isothermal expansion or compression. The change in internal energy in this case is $\inline \Delta U = 0$ , as $\inline T_1=T_2$ (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:

$Q = W$

(48)

Adiabatic Expansion Or Compression

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

$Q = 0$

(49)

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

$PdV = -dU$

(50)

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

$dU = m C_V dT$

(51)

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

$PdV = - m C_V dT$

(52)

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

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

(53)

or, furthermore, as:

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

(54)

By integrating equation (54) we get that:

$R ln V + C_V ln T = const.$

(55)

which can also be written as:

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

(56)

or, furthermore, as:

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

(57)

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

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

(58)

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

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

(59)

From equation (36) we can also write that:

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

(60)

Therefore, equation (59) becomes:

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

(61)

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

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

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

(62)

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

$W = -m C_V (T_2 - T_1)$

(63)

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

$W = -\Delta U$

(64)

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

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

$W = \Delta U$

(65)

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