# Why Define Internal Energy Using Average Mechanical Energy?

• member 731016
In summary, the conversation discusses the definition of internal energy and how it is defined differently in the Pink equation. The individuals also mention a potential error in the textbook they are using and suggest finding a better book or using common sense to eliminate mistakes in trivial definitions. The textbook in question is the OpenStax physics volume 2. It is noted that OpenStax has a history of errors and there is a discussion about the potential work needed to bring a molecule from infinity to its present location.
member 731016
Homework Statement
Relevant Equations
For this,

They say internal energy is the sum of the all the mechanical energies of each particle in within the thermodynamic system, however, they then define internal energy differently using the average mechanical energy for all particles within the system (Pink equation). Does someone please know why they did that?

Many thanks!

ChiralSuperfields said:
Does someone please know why they did that?
No, nor why they talk about the "bars over K and U" when there clearly aren't any. You have two choices:
1. find a better book; or
2. use your common sense to eliminate mistakes in trivial definitions.
Posting here to ask us why an unknown author makes mistakes won't help anyone.

member 731016
pbuk said:
No, nor why they talk about the "bars over K and U" when there clearly aren't any. You have two choices:
1. find a better book; or
2. use your common sense to eliminate mistakes in trivial definitions.
Posting here to ask us why an unknown author makes mistakes won't help anyone.

There is actually bars,

I think the highlighter might have made them hard to see.

Sorry I don't have any common sense since I am do not have any other experience with an equation of internal energy. What internal energy equation do you use?

Many thanks!

It looks to me like a conflation of two expressions: ##E_{int}=\Sigma_{i=1}^N(K_i+U_i)=N(\bar K+\bar U)##.
But the opening sentence is wrong, it is not the sum of individual kinetic and potential energies.
For the KE, as is later clarified, the sum is over the KEs in the frame of reference of the common mass centre.
For potential energy, the sum over i makes no sense since individual molecules do not have PE. Internal PE resides in the forces between the molecules and the potential of those forces to do work. Further, this is distinct from PE involving external forces.

What is the book?

Chestermiller, pbuk and member 731016
haruspex said:
It looks to me like a conflation of two expressions: ##E_{int}=\Sigma_{i=1}^N(K_i+U_i)=N(\bar K+\bar U)##.
But the opening sentence is wrong, it is not the sum of individual kinetic and potential energies.
For the KE, as is later clarified, the sum is over the KEs in the frame of reference of the common mass centre.
For potential energy, the sum over i makes no sense since individual molecules do not have PE. Internal PE resides in the forces between the molecules and the potential of those forces to do work. Further, this is distinct from PE involving external forces.

What is the book?

The textbook is the OpenStax physics volume 2. Here is a link to the section I am referring to: https://openstax.org/books/university-physics-volume-2/pages/3-2-work-heat-and-internal-energy

Many thanks!

Sadly that is the textbook we are using for my course.

scottdave, MatinSAR, malawi_glenn and 2 others
haruspex said:
Oh dear. OpenStax openly sux. I have reported hundreds of errors to them, many of which they refuse to fix.
Thank you for your reply @haruspex! Yeah there is a quite a few errors :(

haruspex said:
It looks to me like a conflation of two expressions: ##E_{int}=\Sigma_{i=1}^N(K_i+U_i)=N(\bar K+\bar U)##.
But the opening sentence is wrong, it is not the sum of individual kinetic and potential energies.
For the KE, as is later clarified, the sum is over the KEs in the frame of reference of the common mass centre.
For potential energy, the sum over i makes no sense since individual molecules do not have PE. Internal PE resides in the forces between the molecules and the potential of those forces to do work. Further, this is distinct from PE involving external forces.

What is the book?
Couldn't ##U_i## be considered the amount of work needed to bring a given molecule from infinity to its present location, holding the locations of all other molecules at their present positions?

member 731016
Chestermiller said:
Couldn't ##U_i## be considered the amount of work needed to bring a given molecule from infinity to its present location, holding the locations of all other molecules at their present positions?
Yes, provided you make it ##\frac 12\Sigma U_i##.

member 731016

## What is the internal energy contradiction?

The internal energy contradiction refers to apparent inconsistencies or misunderstandings in the application of the concept of internal energy in thermodynamics. This often stems from confusion about how internal energy is defined, calculated, or interpreted in different contexts, such as closed versus open systems or during various thermodynamic processes.

## How does the internal energy contradiction arise in thermodynamic processes?

The contradiction can arise when there is a misunderstanding of the first law of thermodynamics, which states that the change in internal energy of a system is equal to the heat added to the system minus the work done by the system. Misinterpretations can occur when differentiating between heat and work interactions, especially in complex processes where multiple forms of energy transfer are involved.

## Can internal energy be directly measured, and how does this relate to the contradiction?

Internal energy cannot be directly measured; it is a state function that can only be inferred from other measurable quantities such as temperature, pressure, and volume. The contradiction often arises because people might expect a direct measurement or might misinterpret indirect measurements, leading to confusion about the actual internal energy of a system.

## How does the concept of internal energy apply to open systems, and what contradictions might this cause?

In open systems, internal energy must account for mass transfer in addition to heat and work interactions. This can lead to contradictions if one does not properly include the energy associated with the mass entering or leaving the system. Failing to account for these factors can lead to incorrect conclusions about the changes in internal energy.

## What are common misconceptions that lead to the internal energy contradiction?

Common misconceptions include confusing internal energy with other forms of energy such as kinetic or potential energy, misunderstanding the role of work and heat in energy transfer, and neglecting the effects of phase changes or chemical reactions. These misconceptions can lead to incorrect applications of thermodynamic principles and the perception of contradictions.

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