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Second law of Thermodynamics
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The second law of thermodynamics states that in any energy conversion process, some of the energy will always be lost as waste heat, and it is impossible to completely convert heat into useful work without any losses. The portion of heat that cannot be converted into work is released as waste heat. This waste heat increases the entropy (a measure of disorder) of the system and its surroundings, contributing to the overall increase in entropy.
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Mechanisms
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The second law of thermodynamics states that in any natural process, the total entropy of an isolated system can never decrease over time; it can only stay constant or increase. Entropy is a measure of disorder or randomness, and the second law implies that natural processes tend to move towards a state of maximum entropy or disorder. This law explains why certain processes are irreversible and why energy tends to spread out or disperse. It also underpins the concept of the arrow of time, giving a direction to time's flow based on the progression towards greater disorder. In practical terms, the second law dictates that no heat engine can be perfectly efficient, as some energy will always be lost as waste heat, and it sets the fundamental limits on the efficiency of energy conversion and the operation of heat engines, refrigerators, and other systems.
Mechanisms
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Heat and entropy
Concept
The first law of thermodynamics states that energy is conserved, and in particular, there are two ways to modify the internal energy of the system, known as the internal energy ($U$). This can be achieved either by adding or removing the caloric Content ($Q$) and by doing work on the system or allowing the system to do work, represented by the effective work ($W$).
The second law restricts these processes, limiting the conversion of the internal energy ($U$) and the effective work ($W$). In this regard, it establishes that it is not possible for all the energy the internal energy differential ($dU$) to be completely converted into useful work the differential inexact labour ($\delta W$), meaning that the differential inexact Heat ($\delta Q$) can never be zero. In other words, it is impossible to convert internal energy into mechanical work without experiencing a loss in the form of heat (the differential inexact Heat ($\delta Q$)).
A second consequence of the second law is that it becomes necessary to introduce a new variable, which serves the role of the volume ($V$) for the effective work ($W$), taking into account that the caloric Content ($Q$) plays its role as the receiver of unutilized energy for work creation. This new variable is called the entropy ($S$), and the third law requires that its variation ( ($$)) is always positive or zero but never negative.
In a system, a subsystem may experience a decrease in entropy ($\Delta S_{sub}<0$), but the entire system must either maintain constant entropy or experience an increase in entropy ($\Delta S_{total}\geq 0$), in accordance with the third law.
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Second law of thermodynamics
Concept
The second law of thermodynamics is formulated from several publications [1,2], establishing that it is not possible to completely convert the energy into useful work. The difference between these quantities relates to the unusable energy the differential inexact Heat ($\delta Q$), which corresponds to the heat generated or absorbed in the process the absolute temperature ($T$).
In the case of the differential inexact labour ($\delta W$), there is a relationship between the intensive variable the pressure ($p$) and the extensive variable the volume ($V$), expressed as:
| $ \delta W = p dV $ |
An intensive variable is characterized by defining the state of the system without depending on its size. In this sense, the pressure ($p$) is an intensive variable, as it describes the state of a system regardless of its size. On the other hand, an extensive variable, such as the volume ($V$), increases with the size of the system.
In the case of the differential inexact Heat ($\delta Q$), an additional extensive variable is needed to complement the intensive variable the absolute temperature ($T$) to define the relationship as follows:
| $ \delta Q = T dS $ |
This new variable, which we will call the entropy ($S$), is presented here in its differential form (the entropy variation ($dS$)) and models the effect that not all energy the internal energy differential ($dU$) can be completely converted into useful work the differential inexact labour ($\delta W$).
[1] "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 That Can Be Derived from It for the Theory of Heat Itself), Rudolf Clausius, Annalen der Physik, 1850
[2] "On the Dynamical Theory of Heat," William Thomson (Lord Kelvin), Transactions of the Royal Society of Edinburgh, 1851
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First law of Thermodynamics and Pressure
Concept
Since the internal energy differential ($dU$) relates to the differential inexact Heat ($\delta Q$) and the differential inexact labour ($\delta W$) as shown below:
| $ dU = \delta Q - \delta W $ |
And it is known that the differential inexact labour ($\delta W$) is related to the pressure ($p$) and the volume Variation ($dV$) as follows:
| $ \delta W = p dV $ |
Therefore, we can conclude that:
| $ dU = \delta Q - p dV $ |
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Internal energy: differential ratio
Concept
Given that the internal energy ($U$) depends on the entropy ($S$) and the volume ($V$), the internal energy differential ($dU$) can be calculated as follows:
$dU = \left(\displaystyle\frac{\partial U}{\partial S}\right)_V dS + \left(\displaystyle\frac{\partial U}{\partial V}\right)_S dV$
To simplify the notation of this expression, we introduce the derivative of the internal energy ($U$) with respect to the entropy ($S$) while keeping the volume ($V$) constant as:
$DU_{S,V} \equiv \left(\displaystyle\frac{\partial U}{\partial S}\right)_V$
and the derivative of the internal energy ($U$) with respect to the volume ($V$) while keeping the entropy ($S$) constant as:
$DU_{V,S} \equiv \left(\displaystyle\frac{\partial U}{\partial V}\right)_S$
therefore, we can write:
| $ dU = DU_{S,V} dS + DU_{V,S} dV $ |
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Model
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Equations
$ \delta Q = T dS $
dQ = T * dS
$ dU = \delta Q - p dV $
dU = dQ - p * dV
$ dU = T dS - p dV $
dU = T * dS - p * dV
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Second law of thermodynamics
Equation
The differential inexact Heat ($\delta Q$) is equal to the absolute temperature ($T$) times the entropy variation ($dS$):
| $ \delta Q = T dS $ |
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First law of thermodynamics and pressure
Equation
With the first law of thermodynamics, it can be expressed in terms of the internal energy differential ($dU$), the differential inexact Heat ($\delta Q$), the pressure ($p$), and the volume Variation ($dV$) as:
| $ dU = \delta Q - p dV $ |
Since the internal energy differential ($dU$) relates to the differential inexact Heat ($\delta Q$) and the differential inexact labour ($\delta W$) as shown below:
| $ dU = \delta Q - \delta W $ |
And it is known that the differential inexact labour ($\delta W$) is related to the pressure ($p$) and the volume Variation ($dV$) as follows:
| $ \delta W = p dV $ |
Therefore, we can conclude that:
| $ dU = \delta Q - p dV $ |
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Internal Energy: differential ratio
Equation
The dependency of the internal energy differential ($dU$) on the pressure ($p$) and the volume Variation ($dV$), in addition to the absolute temperature ($T$) and the entropy variation ($dS$), is given by:
| $ dU = T dS - p dV $ |
As the internal energy differential ($dU$) depends on the differential inexact Heat ($\delta Q$), the pressure ($p$), and the volume Variation ($dV$) according to the equation:
| $ dU = \delta Q - p dV $ |
and the expression for the second law of thermodynamics with the absolute temperature ($T$) and the entropy variation ($dS$) as:
| $ \delta Q = T dS $ |
we can conclude that:
| $ dU = T dS - p dV $ |
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