Expansion and Compression of Ideal Gases

Key facts

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

$PV^n = const.$

(1)

where $n$ is the polytropic index.

The work done during a polytropic expansion is given by:

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

(2)

or by:

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

(3)

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

The work done during a polytropic compression is given by:

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

(4)

or by:

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

(5)

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

The heat supplied during a polytropic expansion is given by:

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

(6)

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

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

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

(7)

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

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

$Q = W$

(8)

where $W$ is the work done.

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

$W = - \Delta U$

(9)

and for an adiabatic compression by:

$W = \Delta U$

(10)

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

<br/>

Constants

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

(11)

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

(12)

A discussion on the expansion and compression of ideal gases, also considering the particular cases of isothermal and adiabatic processes

You're viewing an older version of this page (#3634). View the current version.

View versions (4)

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

$PV^n = const.$

(13)

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:

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

(14)

Therefore, when dividing equation (#1) by $\displaystyle \frac{PV}{T}$ we will also get a constant:

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

(15)

or, written in a different form:

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

(16)

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

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

(17)

Also, we can rewrite equation (#2) as:

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

(18)

By using the expression form of the volume from (#6) in equation (#1), we get that:

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

(19)

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

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

(20)

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:

$W = \int_1^2 PdV$

(21)

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

$PV^n = C$

(22)

or, furthermore, that:

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

(23)

where $C$ is a constant, and $n$ the polytropic index. By using the expression of pressure from (#11) in equation (#9), we get the work done by the gas as:

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

(24)

which, integrated, leads to:

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

(25)

By using identity (#10) again, we can rewrite (#13) as:

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

(26)

or, furthermore, as:

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

(27)

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

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

(28)

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

$PV = m \tilde{R} T$

(29)

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

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

(30)

and:

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

(31)

By using (#18) and (#19), the work done from equation (#16) becomes:

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

(32)

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

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

(33)

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:

$W = - \int_1^2 PdV$

(34)

Therefore, equations (#16) and (#21) can be rewritten in order to express the work done during a polytropic compression as:

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

(35)

and:

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

(36)

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

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

(37)

or as:

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

(38)

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 $W$ :

$Q = \Delta U + W$

(39)

As the change in internal energy is given by:

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

(40)

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

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

(41)

which can also be written as:

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

(42)

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:

$C_P - C_V = \tilde{R}$

(43)

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

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

(44)

we also have that:

$C_P = \gamma C_V$

(45)

By using the expression of $C_P$ from (#33) in equation (#31), we get that:

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

(46)

or, furthermore, that:

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

(47)

from which we obtain $C_V$ as:

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

(48)

By using the expression of $C_V$ from (#36) in equation (#30), we get the heat supplied during a polytropic expansion as:

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

(49)

or, furthermore, as:

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

(50)

Equation (#38) leads to:

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

(51)

or, furthermore, to:

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

(52)

Taking into account that $\displaystyle \frac{m \tilde{R} (T_2 - T_1)}{1-n} = W$ (see equation #21), we obtain the heat supplied during a polytropic expansion as:

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

(53)

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 (#1) will be equivalent to other particular processes. For instance, if $n=1$ , equation (#1) will be rewritten as:

$PV = const.$

(54)

However, taking into account the ideal gas law (see #17), this also means that:

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

(55)

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

$T = const.$

(56)

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

As the work done during a process which changes the system from state $1$ to state $2$ is given by $W=\int_1^2 PdV$ , and also taking into account that $\displaystyle P=\frac{m \tilde{R} T}{V}$ (see the ideal gas law, equation #17), we get the work done during an isothermal expansion or compression as:

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

(57)

from which we obtain:

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

(58)

Let us now consider the heat changed 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 #28). Therefore, from the first law of thermodynamics (see #27), we obtain that, for an isothermal process, the heat changed equals the work done:

$Q = W$

(59)

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 changed during the process (see equation #41) will be:

$Q = 0$

(60)

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 #27) written for infinitesimal changes, that:

$PdV = -dU$

(61)

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

$dU = m C_V dT$

(62)

where $m$ is the mass, $C_V$ the heat capacity at constant volume, and $dT$ the infinitesimal change in temperature. Therefore, equation (#49) becomes:

$PdV = - m C_V dT$

(63)

However, $\displaystyle P=\frac{m \tilde{R} T}{V}$ (see the ideal gas law, equation #17). Hence, we can also write equation (#51) as:

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

(64)

or, furthermore, as:

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

(65)

By integrating equation (#53) we get that:

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

(66)

which can also be written as:

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

(67)

or, furthermore, as:

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

(68)

By raising $e$ to equation (#56), we get that:

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

(69)

Then, by raising equation (#57) to the $\displaystyle \frac{1}{C_V}$ power, we get that:

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

(70)

From equation (#35) we can also write that:

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

(71)

Therefore, equation (#58) becomes:

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

(72)

However, we know that all polytropic processes satisfy $V^{n-1} \cdot T = const.$ (see equation #5). By coupling this with equation (#60) 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 #21):