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Adiabatic cooling
Storyboard 
As the air rises it reaches areas of lower pressure so it begins to decompress. As this happens at a relatively high speed the gas must perform the necessary work with the energy it has without being able to absorb it from the outside. This leads to a cooling called decompression or adiabatic cooling.
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Adiabatic Curves
Image
The three adiabatic curves are shown below:
* pressure vs volume
* pressure vs temperature
* volume vs temperature
To compare the adiabatic curves are shown next to their corresponding curves from the equation of ideal cases.
Note the large difference between the volume vs. temperature curve and the same relationship in the isobaric case. This means that only in the case that the air expands under adiabatic conditions does a temperature reduction occur. In the isobaric case the opposite occurs.
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Adiabatic process
Note
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] "Réflexions sur la puissance motrice du feu" (Reflections on the Motive Power of Fire), Sadi Carnot, 1824
[2] "Über die bewegende Kraft der Wärme und die Gesetze, welche sich daraus für die Wärmelehre 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
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Adiabatic cooling
Storyboard
As the air rises it reaches areas of lower pressure so it begins to decompress. As this happens at a relatively high speed the gas must perform the necessary work with the energy it has without being able to absorb it from the outside. This leads to a cooling called decompression or adiabatic cooling.
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Equations
The pressure ($p$), the volume ($V$), the absolute temperature ($T$), and the number of moles ($n$) are related through the following physical laws:
• Boyle's law
• Charles's law
• Gay-Lussac's law
• Avogadro's law
These laws can be expressed in a more general form as:
$\displaystyle\frac{pV}{nT}=cte$
This general relationship states that the product of pressure and volume divided by the number of moles and temperature remains constant:
When the pressure ($p$) behaves as an ideal gas, satisfying the volume ($V$), the number of moles ($n$), the absolute temperature ($T$), and the universal gas constant ($R_C$), the ideal gas equation:
and the definition of the molar concentration ($c_m$):
lead to the following relationship:
Since with the variation of the internal energy ($dU$), the variation of heat ($\delta Q$), and the differential inexact labour ($\delta W$) we have:
$dU = \delta Q - \delta W = 0$
We can replace the variation of heat ($\delta Q$) with the infinitesimal version of the equation for the heat supplied to liquid or solid ($\Delta Q$) involving the specific heat at constant pressure ($c_p$), the mass ($M$), and the temperature variation ($\Delta T$) in the case of constant pressure, as shown below:
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:
$c_pMdT=-pdV$
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:
In the adiabatic case, for ERROR:5177,0 and the volume ($V$) with the universal gas constant ($R_C$), the molar Mass ($M_m$), the specific heat at constant pressure ($c_p$), 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:
With the values of the volume in state i ($V_i$), the volume in state f ($V_f$), the temperature in initial state ($T_i$), the temperature in final state ($T_f$), and the adiabatic index ($\kappa$), the following relationship is established:
By utilizing the gas equation with the parameters the pressure ($p$), the volume ($V$), the number of moles ($n$), the universal gas constant ($R_C$), and the absolute temperature ($T$), we derive the following expression:
This equation describes how, in an adiabatic process varying from an initial situation to a final one in terms of the pressure ($p$) and the absolute temperature ($T$), it relates to the pressure in initial state ($p_i$) and the pressure in final state ($p_f$) as follows:
With the values the volume in state i ($V_i$), the volume in state f ($V_f$), the temperature in initial state ($T_i$), the temperature in final state ($T_f$), and the adiabatic index ($\kappa$), the following relationship is presented:
Using the gas equation with the parameters the pressure ($p$), the volume ($V$), the number of moles ($n$), the universal gas constant ($R_C$), and the absolute temperature ($T$), we obtain the following expression:
This equation describes how, in an adiabatic process that varies from an initial situation to a final one in terms of the pressure ($p$) and the volume ($V$), it is related to the pressure in initial state ($p_i$) and the pressure in final state ($p_f$) as follows:
Examples
The internal energy differential ($dU$) is always equal to the amount of the differential inexact Heat ($\delta Q$) supplied to the system (positive) minus the amount of the differential inexact labour ($\delta W$) performed by the system (negative):
In the adiabatic case, the system does not have the ability to alter the caloric Content ($Q$), meaning that the differential inexact Heat ($\delta Q$) must be zero:
In the adiabatic case it is given that the universal gas constant ($R_C$), the molar Mass ($M_m$), and the specific heat of gases at constant volume ($c_V$) vary in the temperature variation ($dT$) and ERROR:5223< /var> according to:
Using 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 ($\Delta V$), the adiabatic index ($\kappa$) can be defined as follows:
The pressure ($p$), the volume ($V$), the absolute temperature ($T$), and the number of moles ($n$) are related by the following equation:
where the universal gas constant ($R_C$) has a value of 8.314 J/K mol.
The molar concentration ($c_m$) corresponds to ERROR:9339,0 divided by the volume ($V$) of a gas and is calculated as follows:
The pressure ($p$) can be calculated from the molar concentration ($c_m$) using the absolute temperature ($T$), and the universal gas constant ($R_C$) as follows:
From an initial state (i) with the volume in state i ($V_i$) and the temperature in initial state ($T_i$) it goes to a final state (f) with the volume in state f ($V_f$) and the temperature in final state ($T_f$) according to:
From an initial state (i) with the pressure in initial state ($p_i$) and the temperature in initial state ($T_i$) it goes to a final state (f) with the pressure in final state ($p_f$) and the temperature in final state ($T_f$) according to:
From an initial state (i) with the pressure in final state ($p_f$) and the volume in state i ($V_i$) it goes to a final state (f) with the pressure in final state ($p_f$) and the volume in state f ($V_f$) according to:
The three adiabatic curves are shown below:
* pressure vs volume
* pressure vs temperature
* volume vs temperature
To compare the adiabatic curves are shown next to their corresponding curves from the equation of ideal cases.
Note the large difference between the volume vs. temperature curve and the same relationship in the isobaric case. This means that only in the case that the air expands under adiabatic conditions does a temperature reduction occur. In the isobaric case the opposite occurs.
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
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