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

[saminator910]

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

$dU = \delta Q + \delta W$

Them the equation for $\delta Q$ :

$\delta Q = TdS$

I just don't get when I should be inputting the $\delta$, and when to put in the d.

 

[Andrew Mason]

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

$dU = \delta Q + \delta W$

Them the equation for $\delta Q$ :

$\delta Q = TdS$

I just don't get when I should be inputting the $\delta$, 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

 

[saminator910]

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?

 

[Andrew Mason]

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

 

[PJC01]

The use of \delta and d to represent infinitesimal changes in equations is a common practice in mathematics and science. Both symbols are used to represent small, incremental changes in a variable, but they may have slightly different meanings depending on the context.

In thermodynamics, the lowercase delta, \delta, is typically used to represent a change in a variable at constant conditions, while the lowercase d, d, is used to represent a small change in a variable that is not necessarily constant. For example, in the equation for change in internal energy, \delta Q + \delta W, the \delta represents small changes in heat and work at constant conditions.

On the other hand, in the equation for change in heat, \delta Q = TdS, the d represents a small change in entropy, which may not necessarily be at constant conditions. This distinction is important because in thermodynamics, the value of a variable can vary depending on the conditions, so it is important to specify whether the change is occurring at constant conditions or not.

In general, the use of \delta and d can vary depending on the specific equation and its context. It is important to carefully read and understand the equations in order to determine which symbol is appropriate to use.