# Thermodynamic functions and state variables

• I
• cianfa72
In summary: Mathematical intuition is behind all the many Legendre transformations you do to transform from one thermodynamic potential to the other.
cianfa72
Hi,

a basic doubt about thermodynamic functions and state variables. Take for instance transformations I and II in the following ##(p,V)## plane.

As far as I can tell, just because the transformations are drawn as continuous lines they are reversible by definition. Namely we can transform the system back in the other way and from a mathematical point of view it really makes sense (e.g. work and heat exchanged with the system's surrounding are the same as in the 'direct' transformation just with sign flipped).

By the way, given a system (e.g. a gas in a cylinder), there exist a pair of state variables for each way the system can exchange energy with the surrounding. In the above case the system can do work by means of gas thermodynamic pressure ##p## and exchange heat with the surrounding. Hence we have 2 pairs: ##(p,V)## and ##(t,S)## respectively.

Now my point is: how many are the independent state variables ?

The 'state equation' for the system involves the variables ##p,V,t## (take for instance the state equation of ideal gas) and I believe there exist another independent equation that involve the entropy ##S##. Thus in that case (system = gas in a cylinder) the independent state variables are just 2.

Does it makes sense ? Thank you.

Note: In the above I use lower case for intensive state variables and upper case the extensive ones.

Last edited:
If you are talking about a single phase gas of constant composition and believe the Phase Rule, it tells us that there are two independent state variables.

Chestermiller said:
If you are talking about a single phase gas of constant composition and believe the Phase Rule, it tells us that there are two independent state variables.
Yes, assume a single phase gas of constant composition (and believe in the Phase Rule).

Sorry, I didn't get your point: is the Phase Rule actually another 'state equation' relating the system's state variables ##p,V,t,S## ?

cianfa72 said:
Yes, assume a single phase gas of constant composition (and believe in the Phase Rule).

Sorry, I didn't get your point: is the Phase Rule actually another 'state equation' relating the system's state variables ##p,V,t,S## ?

From a mathematical point of view, we can employ the 'contact manifold' geometry to represent a thermodynamic system (material/substance). Basically it is a 5D manifold endowed with a contact structure -- Contact geometry

This contact structure represents the thermodynamic first law:

$$dU - TdS - pdV = 0$$
##d\omega = dU - TdS - pdV## is the one-form contact structure defined on the 5D manifold. Each material/substance is actually represented by a 2D submanifold of it.

Basically the constrains between the 5 state variables ##(U,S,V,T,p)## define such submanifold and some of them are actually the 'state equations' for the given substance.

See the following:
thermodynamic formalism
MathThermoStates

vanhees71
cianfa72 said:
From a mathematical point of view, we can employ the 'contact manifold' geometry to represent a thermodynamic system (material/substance). Basically it is a 5D manifold endowed with a contact structure -- Contact geometry

This contact structure represents the thermodynamic first law:

$$dU - TdS - pdV = 0$$
##d\omega = dU - TdS - pdV## is the one-form contact structure defined on the 5D manifold. Each material/substance is actually represented by a 2D submanifold of it.

Basically the constrains between the 5 state variables ##(U,S,V,T,p)## define such submanifold and some of them are actually the 'state equations' for the given substance.

See the following:
thermodynamic formalism
MathThermoStates
I have no idea what any of this means, and, in my scheme of things (focused on solving actual problems), it doesn't matter.

Yes, mine was a theoretical approach to address the following: taking for instance the 'ideal gas substance' why its state can be described by just 2 state variables picked from the set of 5 state variables above ?

$$\mathrm{d} U-T \mathrm{d} S + P \mathrm{d} V=0.$$
Otherwise it's just the geometric intuition behind all the many Legendre transformations you do to transform from one thermodynamic potential to the other.

vanhees71 said:
$$\mathrm{d} U-T \mathrm{d} S + P \mathrm{d} V=0.$$
Oops, yes of course.

vanhees71

## 1. What are thermodynamic functions?

Thermodynamic functions are mathematical quantities that describe the behavior of a thermodynamic system. They include properties such as energy, temperature, pressure, and volume, and can be used to calculate changes in the system as it undergoes various processes.

## 2. What are state variables in thermodynamics?

State variables are properties of a thermodynamic system that are independent of the path taken to reach a particular state. These include temperature, pressure, and volume, and are used to describe the current state of a system.

## 3. How do thermodynamic functions relate to state variables?

Thermodynamic functions are functions of state variables, meaning that they depend on the values of state variables to calculate the behavior of a system. For example, the internal energy of a system is a function of its temperature and volume.

## 4. What are the four primary thermodynamic functions?

The four primary thermodynamic functions are internal energy, enthalpy, entropy, and free energy. These functions are used to describe the behavior of a system and can be calculated using state variables.

## 5. How are thermodynamic functions and state variables used in practical applications?

Thermodynamic functions and state variables are used in a variety of practical applications, including the design of engines, refrigeration systems, and chemical reactions. They allow scientists and engineers to predict and control the behavior of thermodynamic systems, making them essential in many industries.

Replies
9
Views
1K
Replies
3
Views
1K
Replies
15
Views
2K
Replies
4
Views
2K
Replies
1
Views
1K
Replies
12
Views
2K
Replies
1
Views
701
Replies
3
Views
1K
Replies
65
Views
75K
Replies
5
Views
2K