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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 first law of thermodynamics states that energy is conserved, meaning that the internal energy differential ($dU$) is always equal to the differential inexact Heat ($\delta Q$) supplied to the system (positive) minus the differential inexact labour ($\delta W$) done by the system (negative).
Therefore, we have:
 $ dU = \delta Q - \delta W $ |
While the exact differential does not depend on how the variation is executed, the inexact differential does. When referring to a differential without specifying that it is inexact, it is assumed to be exact.
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Adiabatic condition
Equation
In the adiabatic case, the system does not have the ability to alter the internal energy ($U$), meaning that the variation of the Internal Energy ($dU$) must be zero:
| $ dU =0$ |
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Temperature and Volume Variation
Equation
From the condition of adiabatic processes in which the internal energy is zero, i.e., the variation of the Internal Energy ($dU$) equals zero, it follows that the variation of heat ($\delta Q$) is equal to the differential inexact labour ($\delta W$), as shown in the following equation:
$dU = \delta Q - \delta W = 0$
If we substitute the variation of heat ($\delta Q$) and the differential inexact labour ($\delta W$) with the corresponding expressions in terms of absolute temperature ($T$) and the volume ($V$), we obtain the equation involving 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 ($dV$):
| $\displaystyle\frac{ dT }{ T }=-\displaystyle\frac{ R }{ M_m c_p }\displaystyle\frac{ dV }{ V }$ |
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 variation of heat ($\Delta Q$) 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:
| $ \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:
| $\displaystyle\frac{ dT }{ T }=-\displaystyle\frac{ R }{ M_m c_p }\displaystyle\frac{ dV }{ V }$ |
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Adiabatic Index
Equation
El indice adiabático se puede calcular mediante:
| $ \kappa =1+\displaystyle\frac{ R }{ M_m c_V }$ |
Para el caso de un gas ideal
$c_p=\displaystyle\frac{5}{2}\displaystyle\frac{R}{M_m}$
por lo que el indice adiabático es igual a
$\kappa=1+\displaystyle\frac{2}{5}=\displaystyle\frac{7}{5}=1.4$
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General Gas Law
Equation
In 1834, Émile Clapeyron recognized that the pressure ($p$), the volume ($V$), the absolute temperature ($T$), and the number of moles ($n$) are related by Boyle's law, Charles's law, Gay-Lussac's law, and Avogadro's law. These laws can be rewritten in the form:
| $ 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_{pV} $ |
• Charles's law
| $\displaystyle\frac{ V }{ T } = C_{VT}$ |
• Gay-Lussac's law
| $\displaystyle\frac{ p }{ T } = C_{pT}$ |
• Avogadro's law
| $\displaystyle\frac{ n }{ V } = C_{nV} $ |
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 $ |
In this equation, the universal gas constant ($R$) assumes the value 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
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 $ |
can be rewritten in terms of the molar concentration ($c_m$) as:
| $ 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
Since the equation for the adiabatic process in 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 ($dV$) is as follows:
| $\displaystyle\frac{ dT }{ T }=-\displaystyle\frac{ R }{ M_m c_p }\displaystyle\frac{ dV }{ V }$ |
We can replace the constants with the adiabatic index ($\kappa$) and integrate over absolute temperature ($T$) and the volume ($V$) from the volume in state i ($V_i$), the volume in state f ($V_f$), the temperature in initial state ($T_i$), and the temperature in final state ($T_f$) to obtain:
| $ 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:
| $\displaystyle\frac{ dT }{ T }=-\displaystyle\frac{ R }{ M_m c_p }\displaystyle\frac{ dV }{ V }$ |
By introducing the adiabatic index ($\kappa$), this equation can be expressed as:
| $ \kappa =1+\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}$ |
It's important to note that this equation is an additional constraint to the general equation for ideal gases, as it represents an additional restriction imposed by the condition that the internal energy does not change.
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Adiabatic case relationship of temperature and pressure
Equation
Using the adiabatic equation for 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 ($\kappa$):
| $ T_i V_i ^{ \kappa -1}= T_f V_f ^{ \kappa -1}$ |
and the gas equation, we can establish a relationship that describes how, in an adiabatic process, variables change from an initial situation to a final situation. In the case of the pressure ($p$) and the absolute temperature ($T$), we have the following with the pressure in initial state ($p_i$) and the pressure in final state ($p_f$):
| $ 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 }$ |
.
It's important to note that this equation is an additional constraint to the general equation for ideal gases, as it represents an additional restriction imposed by the condition that the internal energy does not change.
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Adiabatic Pressure and Volume Case Relationship
Equation
Using the adiabatic equation for 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 ($\kappa$):
| $ T_i V_i ^{ \kappa -1}= T_f V_f ^{ \kappa -1}$ |
and the ideal gas equation, we can derive a relationship that describes how, in an adiabatic process, the variables change from an initial situation to a final situation. In the case of the pressure ($p$) and the volume ($V$), we have the following with the pressure in initial state ($p_i$) and the pressure in final state ($p_f$):
| $ 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 }$ |
It's important to note that this equation is an additional constraint to the general equation for ideal gases, as it represents an additional restriction imposed by the condition that the internal energy does not change.
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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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Temperature change in an adiabatic process
Concept
When a gas expands rapidly, the vapor molecules do not have enough time to exchange energy with each other, resulting in a process where there is no change in internal energy, i.e., the variation of the Internal Energy ($dU$) remains constant:
$dU = 0$
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
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