Storyboard

Internal energy is the inherent energy of the system, i.e., the sum of the kinetic and potential energies of the particles comprising it.

Internal energy is a function of the system's state and depends solely on the current state, regardless of how it reached that state.

>Model

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Iframe

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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.

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Concept

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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:

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

we can conclude that:

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Concept

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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 ($dV$), is expressed as:

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

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

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Concept

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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:

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Concept

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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:

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

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

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Concept

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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:

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

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

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Concept

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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 ($T$)

,

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 ($p$)

,

we can conclude that:

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Top

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Parameters

$DU_{S,V}$

DU_SV

Partial derivative of internal energy with respect to entropy at constant volume

K

$DU_{V,S}$

DU_VS

Partial derivative of internal energy with respect to volume at constant entropy

Pa

$Dp_{S,V}$

Dp_SV

Partial derivative of pressure with respect to entropy at constant volume

K/m^3

$DT_{V,S}$

DT_VS

Partial derivative of temperature with respect to volume at constant entropy

K/m^3

Variables

$T$

T

Absolute temperature

K

$S$

S

Entropy

J/K

$dS$

dS

Entropy variation

J/K

$U$

U

Internal energy

J

$dU$

dU

Internal energy differential

J

$p$

p

Pressure

Pa

$dU$

dU

Variation of the internal energy

J

$V$

V

Volume

m^3

$dV$

dV

Volume Variation

m^3

Calculations

• Alternative method
• Traditional method
• Strategy
• 2D
• 3D

Equation

Number

First, select the equation: to , then, select the variable: to

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Calculations

Symbol

Equation

Solved

Translated

Calculations

Symbol

Equation

Solved

Translated

Variable Given Calculate Target : Equation To be used

Equations

ID:(15326, 0)

Equation

>Top, >Model

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 $

Conditions

Variables

$T$

Absolute temperature

$K$

5177

$dS$

Entropy variation

$J/K$

5225

$p$

Pressure

$Pa$

5224

$dU$

Variation of the internal energy

$J$

5400

$dV$

Volume Variation

$m^3$

5223

Relation

Development

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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Equation

>Top, >Model

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

$ U = T S - p V $

Conditions

Variables

$T$

Absolute temperature

$K$

5177

$S$

Entropy

$J/K$

5227

$U$

Internal energy

$J$

5228

$p$

Pressure

$Pa$

5224

$V$

Volume

$m^3$

5226

Relation

Development

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 ($dV$), is expressed as:

$ dU = T dS - p dV $

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

$ U = T S - p V $

ID:(3472, 0)

Equation

>Top, >Model

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 ($p$):

$ DU_{V,S} =- p $

Conditions

Variables

$DU_{V,s}$

Partial derivative of internal energy with respect to volume at constant entropy

$Pa$

8734

$p$

Pressure

$Pa$

5224

Relation

Development

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:

$ dU = DU_{S,V} dS + DU_{V,S} dV $

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

$ dU = T dS - p dV $

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

$ DU_{V,S} =- p $

ID:(3535, 0)

Equation

>Top, >Model

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 ($T$):

$ DU_{S,V} = T $

Conditions

Variables

$T$

Absolute temperature

$K$

5177

$DU_{S,V}$

Partial derivative of internal energy with respect to entropy at constant volume

$K$

8735

Relation

Development

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:

$ dU = DU_{S,V} dS + DU_{V,S} dV $

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

$ dU = T dS - p dV $

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

$ DU_{S,V} = T $

ID:(3546, 0)

Equation

>Top, >Model

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:

$ dU = DU_{S,V} dS + DU_{V,S} dV $

Conditions

Variables

$dS$

Entropy variation

$J/K$

5225

$dU$

Internal energy differential

$J$

8736

$DU_{S,V}$

Partial derivative of internal energy with respect to entropy at constant volume

$K$

8735

$DU_{V,s}$

Partial derivative of internal energy with respect to volume at constant entropy

$Pa$

8734

$dV$

Volume Variation

$m^3$

5223

Relation

Development

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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Equation

>Top, >Model

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

$ DT_{V,S} =- Dp_{S,V} $

Conditions

Variables

$Dp_{S,V}$

Partial derivative of pressure with respect to entropy at constant volume

$K/m^3$

8739

$DT_{V,S}$

Partial derivative of temperature with respect to volume at constant entropy

$K/m^3$

8738

Relation

Development

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 ($T$)

$ DU_{S,V} = T $

,

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 ($p$)

$ DU_{V,S} =- p $

,

we can conclude that:

$ DT_{V,S} =- Dp_{S,V} $

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