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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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First law of thermodynamics
Equation
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):
 $ dU = \delta Q - \delta W $ |
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Adiabatic condition
Equation
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:
| $ \delta Q =0$ |
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Temperature and Volume Variation
Equation
In the adiabatic case it is given that the universal gas constant ($R$), the molar Mass ($M_m$), and the specific heat of gases at constant volume ($c_V$) vary in the temperature variation ($dT$) and ($$)5223< /var> according to:
| $\displaystyle\frac{ dT }{ T }=-\displaystyle\frac{ R }{ M_m c_V }\displaystyle\frac{ dV }{ V }$ |
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_s$) involving the specific heat at constant pressure ($c_p$), the mass ($M$), and the temperature variation in a liquid or solid ($\Delta T_s$) in the case of constant pressure, as shown below:
| $ \Delta Q = c_p M \Delta T $ |
Similarly, we can replace the differential inexact labour ($\delta W$) with the pressure ($p$) and the volume Variation ($dV$):
| $ \delta W = p dV $ |
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$), and número de Moles ($n$), leads to:
| $ p V = n R T $ |
And with the mass ($M$) and the molar Mass ($M_m$):
| $ n = \displaystyle\frac{ M }{ M_m }$ |
Finally, in the limit $\Delta T \rightarrow dt$, we obtain the relationship:
| $\displaystyle\frac{ dT }{ T }=-\displaystyle\frac{ R }{ M_m c_V }\displaystyle\frac{ dV }{ V }$ |
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Adiabatic Index
Equation
Using the universal gas constant ($R$), the molar Mass ($M_m$), the specific heat of gases at constant volume ($c_V$), the temperature variation ($dT$), and the volume Variation ($dV$), the adiabatic index ($\kappa$) can be defined as follows:
| $ \kappa \equiv1+\displaystyle\frac{ R }{ M_m c_V }$ |
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General gas law
Equation
The pressure ($p$), the volume ($V$), the absolute temperature ($T$), and the number of moles ($n$) are related by the following equation:
| $ p V = n R T $ |
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
| $ p V = C_b $ |
• Charles's law
| $\displaystyle\frac{ V }{ T } = C_c$ |
• Gay-Lussac's law
| $\displaystyle\frac{ p }{ T } = C_g$ |
• Avogadro's law
| $\displaystyle\frac{ n }{ V } = C_a $ |
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:
| $ p V = n R T $ |
where the universal gas constant ($R$) has a value of 8.314 J/K·mol.
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Molar concentration
Equation
The molar concentration ($c_m$) corresponds to number of moles ($n$) divided by the volume ($V$) of a gas and is calculated as follows:
| $ c_m \equiv\displaystyle\frac{ n }{ V }$ |
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Pressure as a function of molar concentration
Equation
The pressure ($p$) can be calculated from the molar concentration ($c_m$) using the absolute temperature ($T$), and the universal gas constant ($R$) as follows:
| $ p = c_m R T $ |
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$), the ideal gas equation:
| $ p V = n R T $ |
and the definition of the molar concentration ($c_m$):
| $ c_m \equiv\displaystyle\frac{ n }{ V }$ |
lead to the following relationship:
| $ p = c_m R T $ |
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Adiabatic case relationship of temperature and volume
Equation
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:
| $ T_i V_i ^{ \kappa -1}= T_f V_f ^{ \kappa -1}$ |
In the adiabatic case, for absolute temperature ($T$) and the volume ($V$) with the universal gas constant ($R$), the molar Mass ($M_m$), the specific heat at constant pressure ($c_p$), the temperature variation ($dT$), and the volume Variation ($dV$), we have the following equation:
| $\displaystyle\frac{ dT }{ T }=-\displaystyle\frac{ R }{ M_m c_V }\displaystyle\frac{ dV }{ V }$ |
By introducing the adiabatic index ($\kappa$), this equation can be expressed as:
| $ \kappa \equiv1+\displaystyle\frac{ R }{ M_m c_V }$ |
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:
| $ T_i V_i ^{ \kappa -1}= T_f V_f ^{ \kappa -1}$ |
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Adiabatic case relationship of temperature and pressure
Equation
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:
| $ p_i ^{1- \kappa } T_i ^{ \kappa }= p_f ^{1- \kappa } T_f ^{ \kappa }$ |
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:
| $ T_i V_i ^{ \kappa -1}= T_f V_f ^{ \kappa -1}$ |
By utilizing the gas equation with the parameters the pressure ($p$), the volume ($V$), the number of moles ($n$), the universal gas constant ($R$), and the absolute temperature ($T$), we derive the following expression:
| $ p V = n R T $ |
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:
| $ p_i ^{1- \kappa } T_i ^{ \kappa }= p_f ^{1- \kappa } T_f ^{ \kappa }$ |
.
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Adiabatic Pressure and Volume Case Relationship
Equation
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:
| $ p_i V_i ^{ \kappa }= p_f V_f ^{ \kappa }$ |
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:
| $ T_i V_i ^{ \kappa -1}= T_f V_f ^{ \kappa -1}$ |
Using the gas equation with the parameters the pressure ($p$), the volume ($V$), the number of moles ($n$), the universal gas constant ($R$), and the absolute temperature ($T$), we obtain the following expression:
| $ p V = n R T $ |
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:
| $ p_i V_i ^{ \kappa }= p_f V_f ^{ \kappa }$ |
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Adiabatic Curves
Php
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
Concept
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 ($$)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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