# ￼Description of Adiabatic Expansion and Compression of Gases

## ￼Description of Adiabatic Expansion and Compression of Gases

Igor A. Stepanov^{*}

^{}Institute of Science and Innovative Technologies, Liepaja University, Liela 14, Liepaja, LV3401, Latvia

Corresponding Author Email: istepanov2001@gmail.com

**DOI : ** http://dx.doi.org/10.53709/ CHE.2022.v03i02.006

##### Abstract

Adiabatic expansion of gases can be described by the traditional equation. It is shown that such this equation successfully describes adiabatic compression and expansion of some gases. It is shown that an equation of the mechanocaloric effect can also be used for the description of such compression and expansion. An alternative equation of an adiabatic process is derived which gives a good description for all these gases. The accuracy of this equation is the same one as that of the traditional one.

##### Keywords

**Introduction**

There is the equation of the adiabatic process [1]:

** , (1) **

where g = *C _{P}*/

*C*is the ratio of isobaric and isochoric heat capacity. One can check Eq. (1) experimentally.

_{V }There is a modified Clément-Desormes method for determination heat capacities of gases [3]: “A gas is maintained in a closed bottle, fitted with a stopcock and a manometer, at room temperature *T*1 and at a pressure *P*2 above atmospheric pressure *P*1. When the stopcock is opened, the gas expands almost adiabatically to atmospheric pressure. During this expansion the gas cools from *T*1 to *T*2. Then the stopcock is closed again, and the gas is allowed to return to thermal equilibrium with surroundings. To determine the heat capacity one can measure *P*1, *P*2, and *T*1, and T_{2}”. In Table 1 there are many experimental values of *P*_{i} and *T*_{i} for some gases.

In the notations of Table 1, Eq. (1) will have the following form:

** , (2) **

**Theory**

One can also apply the approach developed in [3]:

During the adiabatic deformation, the enthalpy of the sample changes at D*H*(*S*,*P*) = D*H*(*S*,*T*). It follows that:

. (3)

According to the tables of thermodynamic derivatives [3],

, (4)

where *C _{P}* is the isobaric heat capacity, and a is the thermal expansion coefficient which is 1/

*T*for an ideal gas. And from the same tables:

From these equations, the equation for the mechanocaloric effect follows:

Now let us check its correctness using the data from Table 1 and comparing with Eq. (2). One can assume that in Eq. (6) d*T* = *T*_{2 }– *T*_{1 }and d*P* = *P*_{2 }– *P*_{1}, and calculate *T _{2}/T*

_{1}. Let us derive the equation of an adiabatic process analogously to [1]. During adiabatic expansion or compression

(7)

Here *C*_{A} = *C*_{V} is the adiabatic heat capacity [5]. Adiabatic expansion in theClément-Desormes metod is almost an isobaric process [2], therefore Eq. (7) can be written as:

(8)

If *C*_{A} is independent of temperature, and if one introduces k = *R*/*C*_{A}, one can obtain from Eq. (8)

(9)

And then:

(10)

In the notations of Table 1, Eq. (10) will have the following form:

** , (11) **

Calculations with Eqs (2). (6) and (11) are presented in Table 1. Eq. (11) gives a very good agreement with the experiment.

**Table 1.** Measurements with a modified* *Clément-Desormes method and calculations with Eqs(2). (6) and (11) for some gases heated by compression [2].

Gas | P_{1}, mm Hg | P_{2}, mm Hg | T_{1}, K | T_{2}, K | T_{1}, / T_{2}, | T_{1}, / T_{2},Eq.(2) | T_{1}, / T_{2},Eq.(6) | T_{1}, / T_{2},Eq.(11) |

Air [2] | 760.9 | 825.7 | 287.03 | 280.40 | 1.02364 | 1.02363 | 1.03241 | 1.02363 |

Air [2] | 754.4 | 817.88 | 285.98 | 279.47 | 1.02329 | 1.02335 | 1.03204 | 1.02335 |

O_{2} [2] | 757.6 | 863.91 | 288.11 | 277.63 | 1.03775 | 1.03768 | 1.05070 | 1.03768 |

H_{2}CO_{3} [2] | 756.3 | 845.35 | 284.36 | 277.24 | 1.02568 | 1.02562 | 1.03197 | 1.02562 |

H_{2} [2] | 755.8 | 868.84 | 288.75 | 277.38 | 1.04099 | 1.04087 | 1.05537 | 1.04087 |

Air [2] | 756.25 | 823.35 | 292.99 | 285.98 | 1.02451 | 1.02444 | 1.03342 | 1.02444 |

Air [2] | 752.0 | 814.11 | 293.12 | 286.55 | 1.02293 | 1.02293 | 1.03148 | 1.02293 |

CO_{2} [2] | 756.5 | 831.56 | 293 .12 | 286.85 | 1.02186 | 1.02183 | 1.02744 | 1.02183 |

Air [2] | 886.9 | 761.9 | 288.91 | 276.66 | 1.04428 | 1.04436 | 1.07023 | |

Air [2] | 875.54 | 752.4 | 288.81 | 276.54 | 1.04437 | 1.04426 | 1.07005 |

.

**Conclusions and Discusions**

It is shown that compression an expansion of gases can be successfully described by an equation for the mechanocaloric effect. Formerly we thought that in an isobaric process a change in the enthalpy is equal to the heat of the process. Here the conclusion of [4] is confirmed: in all processes a change in the enthalpy is equal to the heat of the process. An alternative equation of an adiabatic process is derived which also gives a good description of compression and expansion of gases.It is obvious that this alternative equation is not worse than the traditional one.

**Statements and Declarations**

**Competing Interests**

The author declares that he has no competing interests or personal relationships that could have appeared to influence the work reported in this paper

The author has no relevant financial or non-financial interests to disclose.

**References**

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https://doi.org /DOI: 10.24214/jcbsc.C.10.2.05157.

- I. A. Stepanov, Physical Science International Journal (2017) Free online. https://doi.org /DOI: 10.9734/PSIJ/2017/32822
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