Notational question, d vs. delta when denoting an infintesimal change

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Discussion Overview

The discussion revolves around the notational differences between the lowercase delta (\(\delta\)) and the standard \(d\) in thermodynamic equations, specifically regarding their use in denoting infinitesimal changes. Participants explore the implications of these notations in the context of exact and inexact differentials, as well as their applications in thermodynamics.

Discussion Character

  • Conceptual clarification
  • Technical explanation
  • Debate/contested

Main Points Raised

  • Some participants note that \(dU\) represents an exact differential related to defined properties or states, while \(\delta Q\) and \(\delta W\) are inexact differentials that depend on the process involved.
  • One participant explains that inexact differentials indicate quantities that cannot be determined solely by the change in state or property of a system.
  • Another participant seeks clarification on the conceptual distinction, suggesting that \(df\) is thought of as "net distance" while \(\delta f\) represents "total distance."
  • A question is raised about the applicability of standard integration on inexact differentials, with a response indicating that integration can be performed if the path is known.

Areas of Agreement / Disagreement

Participants express varying interpretations of the notational differences, with some agreeing on the definitions of exact and inexact differentials, while others seek further clarification on their applications and implications. The discussion remains unresolved regarding the broader implications of using these notations in different contexts.

Contextual Notes

Participants do not fully resolve the conditions under which standard integration can be applied to inexact differentials, nor do they clarify all assumptions related to the definitions of exact and inexact differentials.

saminator910
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I have seen several thermodynamic equations represented with the lowercase delta, [itex]\delta[/itex], and the standard d to represent an infinitesimal change. For example, the change in internal energy is denoted in Wikipedia as:

[itex]dU = \delta Q + \delta W[/itex]

Them the equation for [itex]\delta Q[/itex] :

[itex]\delta Q = TdS[/itex]

I just don't get when I should be inputting the [itex]\delta[/itex], and when to put in the d.
 
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saminator910 said:
I have seen several thermodynamic equations represented with the lowercase delta, [itex]\delta[/itex], and the standard d to represent an infinitesimal change. For example, the change in internal energy is denoted in Wikipedia as:

[itex]dU = \delta Q + \delta W[/itex]

Them the equation for [itex]\delta Q[/itex] :

[itex]\delta Q = TdS[/itex]

I just don't get when I should be inputting the [itex]\delta[/itex], and when to put in the d.
There are a variety of ways to explain this.

One uses an exact differential when a defined property or state, such as U or S, changes. Q and W are quantities that depend on the process involved in the change in state of a system or surroundings. They do not relate to a change in property or state of a system or surroundings. Since these quantities cannot be determined from the change in state or property of the system or surroundings (eg. dU or dS) we use a different symbol (δQ and δW) which are referred to as inexact differentials.

It is not that the quantities represented by δQ and δW are really inexact for a given process. Rather the inexact differential just indicates that they cannot be determined by knowing the change in state.

AM
 
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Alright, thanks. I read up a little more about those and that makes sense, but I want to make sure I have this straight. So where df is the exact differential, it is though of as "net distance", displacement. On the other hand δf would be inexact, and thought of as the total distance. I have one more question, can I use standard integration on an inexact differential?
 
saminator910 said:
Alright, thanks. I read up a little more about those and that makes sense, but I want to make sure I have this straight. So where df is the exact differential, it is though of as "net distance", displacement. On the other hand δf would be inexact, and thought of as the total distance.
You could represent a change in displacement from an origin in moving from a particular point as an exact differential and the total path distance as an inexact differential.

I have one more question, can I use standard integration on an inexact differential?
If you know the path, you can use integration.

AM
 

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