Internal energy is the total energy contained within a system, including the kinetic energy of molecules moving and vibrating, and the potential energy from the forces between the molecules. It encompasses all the microscopic forms of energy that are not related to the motion or position of the system as a whole, such as thermal energy and chemical energy.

The internal energy of a system changes when heat is added to or removed from the system, or when work is done by or on the system. This is expressed in the first law of thermodynamics, which states that the change in internal energy is equal to the heat added to the system minus the work done by the system.

Internal energy is a state function, meaning it depends only on the current state of the system and not on how the system reached that state. This property allows for the calculation of energy changes between different states using state variables like temperature, pressure, and volume.

mechanisms

As the internal energy differential ($dU$) depends on the differential inexact Heat ($\delta Q$), the pressure ($p$), and the volume Variation ($\Delta V$) according to the equation:

equation=3470

and the expression for the second law of thermodynamics with the absolute temperature ($T$) and the entropy variation ($dS$) as:

equation=9639

we can conclude that:

equation=3471

If the absolute temperature ($T$) and the pressure ($p$) are kept constant, the variation of the internal energy ($dU$), which depends on the entropy variation ($dS$) and the volume Variation ($\Delta V$), is expressed as:

equation=3471

Integrating this results in the following expression in terms of the internal energy ($U$), the entropy ($S$), and the volume ($V$):

equation=3472

[1] " ber die quantitative und qualitative Bestimmung der Kr fte" (On the Quantitative and Qualitative Determination of Forces), Julius Robert von Mayer, Annalen der Chemie und Pharmacie, 1842

[2] " ber die Erhaltung der Kraft" (On the Conservation of Force), Hermann von Helmholtz, 1847

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:

equation=8185

The internal energy differential ($dU$) is a function of the variations in the entropy ($S$) and the volume ($V$), as well as the slopes the partial derivative of internal energy with respect to volume at constant entropy ($DU_{V,S}$) and the partial derivative of internal energy with respect to entropy at constant volume ($DU_{S,V}$), which is expressed as:

equation=8185

When compared with the equation for the internal energy differential ($dU$):

equation=3471

it results in the slope of the internal energy ($U$) with respect to the variation in the volume ($V$):

equation=3535

The internal energy differential ($dU$) is a function of the variations in the entropy ($S$) and the volume ($V$), as well as the slopes the partial derivative of internal energy with respect to volume at constant entropy ($DU_{V,S}$) and the partial derivative of internal energy with respect to entropy at constant volume ($DU_{S,V}$), which is expressed as:

equation=8185

When compared with the equation for the internal energy differential ($dU$):

equation=3471

it results in the slope of the internal energy ($U$) with respect to the variation in the entropy ($S$):

equation=3546

Since the internal energy differential ($dU$) is an exact differential, we should note that the internal energy ($U$) with respect to the entropy ($S$) and the volume ($V$) must be independent of the order in which the function is derived:

$D(DU_{S,V}){V,S}=D(DU{V,S})_{S,V}$

Using the relationship between the slope the partial derivative of internal energy with respect to entropy at constant volume ($DU_{S,V}$) and the absolute temperature ($T$)

equation=3546,

and the relationship between the slope the partial derivative of internal energy with respect to volume at constant entropy ($DU_{V,S}$) and the pressure ($p$)

equation=3535,

we can conclude that:

equation=3556

model

The dependency of the internal energy differential ($dU$) on the pressure ($p$) and the volume Variation ($\Delta V$), in addition to the absolute temperature ($T$) and the entropy variation ($dS$), is given by:

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The internal energy ($U$) is with the absolute temperature ($T$), the pressure ($p$), the entropy ($S$) and the volume ($V$) equal to:

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Comparing this with the first law of thermodynamics, it turns out that the partial derivative of internal energy with respect to volume at constant entropy ($DU_{V,S}$) is equal to minus the pressure ($p$):

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Comparing this with the first law of thermodynamics, it turns out that the partial derivative of internal energy with respect to entropy at constant volume ($DU_{S,V}$) is equal to the absolute temperature ($T$):

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Given that the internal energy ($U$) is a function of the entropy ($S$) and the volume ($V$), the internal energy differential ($dU$) can be expressed as follows:

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With the entropy ($S$), the volume ($V$), the absolute temperature ($T$) and the pressure ($p$) we obtain one of the so-called Maxwell relations:

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