An adiabatic change in a gas occurs when the process is so rapid that there is no time for the system to exchange heat with the surrounding medium. A classic example is the ascent of an air mass in the atmosphere: as it rises to a region of lower pressure and does not interact thermally with its environment, the gas expands. This expansion requires mechanical work, which is performed at the expense of the gas's internal energy, resulting in a decrease in temperature. This final temperature usually differs from the ambient temperature.

The temperature after such a change can be calculated using the adiabatic relations. Once the change occurs, if there is a temperature difference between the gas and the environment, heat exchange begins: the gas absorbs heat if its temperature is lower than the mediums, or releases heat if it is higher. In this stage, the adiabatic equations are no longer valid, and the system evolves toward thermal equilibrium.

You can explore this behavior with the following simulator. Set the initial pressure and temperature of the gas, and the pressure and temperature of the surrounding medium. The simulator will first show the adiabatic change ($\delta Q=0$), followed by the thermal evolution as the gas exchanges heat. The graph shows the heat absorbed (positive) or released (negative):

You may also experiment with different gases by adjusting $\kappa$ (the heat capacity ratio), the molar heat capacity (default: 20.79 J/mol K), and the thermal conductivity (default: 1 J/K), using values that are physically realistic or atmospheric.Note that the ideal gas law (relating pressure $p$, volume $V$, and temperature $T$) always holds. A sudden change in pressure modifies only one of these variables, so an additional equation is needed. In adiabatic conditions, this is provided by the first law of thermodynamics, which simplifies to a relation between $V$ and $T$ when no heat is exchanged. Once heat exchange begins, the process is no longer adiabatic, but the ideal gas law remains valid. The adiabatic equations do not contradict the gas laws; they provide extra constraints that fully determine the system during the rapid transition.

(ID 15262)

When a gas expands rapidly, the water vapor molecules do not have enough time to exchange energy with the surroundings, so no heat is transferred, that is, the variation of heat ($\delta Q$) remains constant:

$\delta Q = 0$

The processes that are carried out under this condition are called adiabatic processes [1,2].

The expansion of the gas requires the system to do work or generate the differential inexact labour ($\delta W$). However, the energy needed for this cannot come from the internal energy ($U$), so it must be obtained from heat. As a result, the temperature of the system decreases, leading to a decrease in the variation of heat ($\delta Q$).

A typical example of this process is the formation of clouds. When air rises through convection, it expands, performs work, and cools down. The moisture in the air condenses, forming clouds.

Conversely, when work is done on the system, positive work the differential inexact labour ($\delta W$) is done. However, since the internal energy ($U$) cannot increase, the thermal energy in the variation of heat ($\delta Q$) increases, leading to an increase in the system's temperature.

A common example of this process is using a pump. If we try to inflate something rapidly, we do work on the system adiabatically, leading to an increase in ERROR:5202 [1] "R flexions sur la puissance motrice du feu" (Reflections on the Motive Power of Fire), Sadi Carnot, 1824 [2] " ber die bewegende Kraft der W rme und die Gesetze, welche sich daraus f r die W rmelehre selbst ableiten lassen" (On the Moving Force of Heat and the Laws Which Can Be Deduced from It for the Theory of Heat Itself), Rudolf Clausius, Annalen der Physik und Chemie, 1850

(ID 41)

The variation of the internal energy ($dU$) in relation to the temperature variation ($\Delta T$) and the heat capacity at constant volume ($C_V$) is expressed as:

Where the heat capacity at constant volume ($C_V$) can be replaced by the specific heat of gases at constant volume ($c_V$) and the mass ($M$) using the following relationship:

Therefore, we obtain:

(ID 15739)

Since with the variation of the internal energy ($dU$), the variation of heat ($\delta Q$), and the differential inexact labour ($\delta W$) we have:

$dU = \delta Q - \delta W = 0 - \delta W = - \delta W$

the variation of the internal energy ($dU$) can be calculated from the specific heat of gases at constant volume ($c_V$), the mass ($M$) and the temperature variation ($\Delta T$) in the case of a constant volume:

Similarly, we can replace the differential inexact labour ($\delta W$) with the pressure ($p$) and the volume Variation ($\Delta V$):

If we equate both expressions, we obtain the equation:

$c_VMdT=-pdV$

Which, with the inclusion of the volume ($V$), the universal gas constant ($R_C$), and ERROR:6679, leads to:

And with the mass ($M$) and the molar Mass ($M_m$):

Finally, in the limit $\Delta T \rightarrow dt$, we obtain the relationship:

(ID 15740)

In the adiabatic case, for the absolute temperature ($T$) and the volume ($V$) with the universal gas constant ($R_C$), the molar Mass ($M_m$), the specific heat of gases at constant volume ($c_V$), the temperature variation ($dT$), and the volume Variation ($\Delta V$), we have the following equation:

By introducing the adiabatic index ($\kappa$), this equation can be expressed as:

This allows us to write the equation as:

$\displaystyle\frac{dT}{T}=-(\kappa - 1)\displaystyle\frac{dV}{V}$

If we integrate this expression between the volume in state i ($V_i$) and the volume in state f ($V_f$), as well as between the temperature in initial state ($T_i$) and the temperature in final state ($T_f$), we obtain:

(ID 15741)

(ID 15321)