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Laws of thermodynamics
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There are three fundamental laws of thermodynamics:
The first law of thermodynamics establishes the conservation of energy, stating that the internal energy of a system is the sum of the heat added to the system minus the work done by the system.
The second law of thermodynamics holds that in an isolated system, the system will evolve towards a state of thermodynamic equilibrium where entropy reaches its maximum value.
The third law of thermodynamics asserts that as the temperature approaches absolute zero, the entropy of the system tends towards a constant value, which is typically zero.
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Mechanisms
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The laws of thermodynamics are fundamental principles that describe how energy is transferred and transformed in physical systems.
The first law, or the law of energy conservation, states that energy cannot be created or destroyed; it can only be transferred or converted from one form to another. This means the total energy of an isolated system remains constant.
The second law states that in any natural process, the total entropy of an isolated system can never decrease over time; it can only stay the same or increase. This principle explains why certain processes are irreversible and why energy tends to spread out or disperse, leading to greater disorder in the system.
The third law asserts that as a system approaches absolute zero, the entropy of the system approaches a minimum value. For a perfectly crystalline substance at absolute zero, the entropy is zero. This law implies that absolute zero cannot be reached through a finite number of processes and establishes a reference point for the determination of entropy.
Together, these laws describe the conservation and dispersion of energy, the direction of spontaneous processes, and the limitations in reaching absolute zero, providing a comprehensive framework for understanding thermodynamic systems.
Mechanisms
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Zero law of thermodynamics
Description
There is a zeroth law of thermodynamics that defines the existence of a transitive property, in the sense that:
• If systems A and B have the same temperature.
• If systems A and C have the same temperature.
Then, necessarily, systems B and C will have the same temperature.
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Energy conservation
Concept
The first law of thermodynamics states that energy is always conserved.
While in mechanics a similar conservation is stated, restricted to non-dissipative systems (for example, excluding friction), thermodynamics generalizes it by considering not only mechanical work but also heat generated or absorbed by the system.
In this sense, the conservation of energy postulated in thermodynamics has no restrictions and is applicable to all systems, as long as all possible energy exchanges and conversions are taken into account.
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First law of thermodynamics
Concept
The first law of thermodynamics was developed through various works [1,2], establishing that energy is conserved. This means 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 we refer to a differential without specifying that it is inexact, it is assumed to be exact.
[1] "Über die quantitative und qualitative Bestimmung der Kräfte" (On the Quantitative and Qualitative Determination of Forces), Julius Robert von Mayer, Annalen der Chemie und Pharmacie, 1842
[2] "Ãœber die Erhaltung der Kraft" (On the Conservation of Force), Hermann von Helmholtz, 1847
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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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Third law of thermodynamics
Description
The third law of thermodynamics [1] states that as the absolute temperature ($T$) of a system approaches absolute zero, the entropy in the zero temperature limit ($S$) of a perfect crystalline substance approaches a minimum value, typically zero. This implies that it is impossible to reach absolute zero through a finite number of processes because entropy would become constant. Essentially, the third law establishes that the entropy of a perfect crystal at absolute zero is zero, and it highlights the unattainability of absolute zero due to the infinite amount of energy removal required
| $\lim_{T\rightarrow 0}S=0$ |
This law has significant implications for understanding the behavior of materials at very low temperatures and provides a fundamental reference point for calculating entropy values.
[1] "Über die Berechnung chemischer Gleichgewichte aus thermischen Messungen" (On the Calculation of Chemical Equilibria from Thermal Measurements), Walther Nernst, Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse, 1906
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Model
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Equations
$ \delta Q = T dS $
dQ = T * dS
$ dU = \delta Q - \delta W $
dU = dQ - dW
$ dU = \delta Q - p dV $
dU = dQ - p * dV
$ dU = T dS - p dV $
dU = T * dS - p * dV
$ \delta W = p dV $
dW = p * dV
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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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Work and pressure
Equation
The differential inexact labour ($\delta W$) is equal to the pressure ($p$) multiplied by the volume Variation ($dV$):
| $ \delta W = p dV $ |
Since the mechanical force ($F$) divided by the section ($S$) is equal to the pressure ($p$):
| $ p \equiv\displaystyle\frac{ F }{ S }$ |
and the volume Variation ($dV$) with the distance traveled ($dx$) is equal to:
| $ \Delta V = S \Delta s $ |
The equation for the differential inexact labour ($\delta W$) can be expressed as:
| $ \delta W = F dx $ |
So it can be written as:
| $ \delta W = p dV $ |
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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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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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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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