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Thermal Balance
Storyboard
Eating allows us to generate energy to perform different activities. However, the process is not efficient, losing a part of the heat energy that the body must eliminate. Elimination occurs by three possible paths; transport, radiation and evaporation.
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Caloric content
Equation
When the variation of heat ($\Delta Q$) is added to a body, we observe a proportional increase of the temperature variation ($\Delta T$). Therefore, we can introduce a proportionality constant the heat capacity ($C$), known as heat capacity, which establishes the following relationship:
 $ \Delta Q = C \Delta T $ |
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General definition of energy
Equation
Carnot was the first to describe energy in terms of the path and the necessary force to traverse it. Progressing along a path with a force requires or generates energy. This corresponds to the equation:
| $ \Delta W = \vec{F} \cdot \Delta\vec{s} $ |
In the continuous limit, the sum can be represented as an integral:
| $ W =\displaystyle\int_C \vec{F} \cdot d \vec{s} $ |
Expanding the concept to a longer path involves summing up the energy required for each path element:
$\bar{W}=\displaystyle\sum_i \vec{F}_i\cdot\Delta\vec{s}_i$
However, the value of this equation represents just an average energy required or generated. The precise energy is obtained when the steps become very small, allowing the force to be considered constant within them:
$W=\displaystyle\sum_i \mbox{lim}_{\Delta\vec{s}_i\rightarrow\vec{0}}\vec{F}_i\cdot\Delta\vec{s}_i$
In this limit, the energy corresponds to the integral along the traveled path, giving us:
| $ W =\displaystyle\int_C \vec{F} \cdot d \vec{s} $ |
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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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Work and pressure
Equation
In mechanics, energy is defined as the product of force and distance traveled. However, when working with gases, it is more practical to use pressure. Since pressure represents force per unit of surface area, it can be shown that the work done, the differential inexact labour ($\delta W$), is equal to pressure, the pressure ($p$), multiplied by the change in volume, 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:
| $ dV = S ds $ |
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 $ |
The change in work is represented by the delta symbol $\delta$ ($\delta W$) instead of the letter d ($dW$), which reminds us that its value depends on the volume change process, $dV$. An example of this would be if the gas undergoes a change during its displacement, in which case:
$\delta W < pdV$
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First law of Thermodynamics and Pressure
Equation
With the first law of thermodynamics, it can be expressed using the internal energy differential ($dU$), the differential inexact Heat ($\delta Q$), and the differential inexact labour ($\delta W$) as:
| $ dU = \delta Q - \delta W $ |
It can be rewritten with work expressed in terms of 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 $ |
This reminds us that heat is an inexact differential. In other words, it matters whether we vary the internal energy first or the volume first when determining the heat the system will produce or absorb.
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Second Law of Thermodynamics
Hypothesis
$dS=\displaystyle\displaystyle\frac{\delta Q}{T}$
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