# Understanding Thermodynamic Variables: Fixing Confusion and Clarifying Relations

• aaaa202
In summary, thermodynamic variables, such as entropy, pressure, temperature, volume, and particle number, can be correlated in certain systems. However, to fully define the state of the system, only a certain number of independent variables are needed, and the others are fixed by the equation of state. This is known as the Gibbs phase rule and can be used to understand relationships between different potentials and their natural independent variables.
aaaa202
So I have had related questions over the past month, but I would like to ask this question to clarify my understanding.
In thermodynamics you work with certain potentials, which are a function of the thermodynamic variables, i.e.:

U(S,T,V,N,P)

Now for U one has the identity:

dU= TdS+SdT-pdV+VdP etc etc.

From these one figure out relations like:

T = dU/dS at fixed V,T,P...

It is this thing about the thermodynamic variables being fixed that has always confused me. In general are the thermodynamic variables S,P,T,V,N not correlated? How am I to understand then the derivative if I am keeping the other thermodynamic variables fixed. Consider for instance including particles of different kinds:

U = ... + μ1N1 + μ2N2

Now we have that:

μ1 = dU/dN1 at fixed T,S,V, N2

But how can I keep N2 fixed if I am in a resevoir, where adding a particle to one phase with N1 particles, actually takes away a particle from the other i.e. dN1=-dN2

Similarly, if I change for instance V, don't I change S or P etc etc.

Usually you use functions with arguments that are independent from each other. In the grand-canonical ensemble there are three independent degrees of freedom.

Different potentials have different "natural" independent variables. For the internal energy you have
$$\mathrm{d} U=T \mathrm{d} S - p \mathrm{d}V + \mu \mathrm{d} N.$$
The natural independent variables for $U$ are thus the entropy, the volume and the particle number, and you have the relations
$$\left (\frac{\partial U}{\partial S} \right )_{V,N}=T, \quad \left (\frac{\partial U}{\partial V} \right )_{S,N}=-p, \quad \left (\frac{\partial U}{\partial N} \right )_{S,V}=\mu.$$
For other combinations of independent variables other potentials are more convenient. E.g., the enthalpy. It's given by a socalled Legendre transformation of the internal energy
$$H=U+p V.$$
$$\mathrm{d} H= \mathrm{d}U + p \mathrm{d} V+V \mathrm{d} p=T \mathrm{d} S + V \mathrm{d} p + \mu \mathrm{d} N.$$
The natural independent variables for the enthalpy are thus the entropy, pressure, and particle number. From this you get
$$\left (\frac{\partial H}{\partial S} \right )_{p,N}=T, \quad \left (\frac{\partial H}{\partial p} \right )_{S,N}=V, \quad \left (\frac{\partial H}{\partial N} \right )_{S,p}=\mu.$$
As you see, it is important to note, which independent variables are to be held fixed when taking a partial derivative.

Other important relations, socalled Maxwell relations, can be found from the 2nd mixed derivatives. E.g., for the internal energy you have
$$\frac{\partial^2 U}{\partial V \partial S}=\left (\frac{\partial T}{\partial V} \right )_{S,N}=-\left (\frac{\partial p}{\partial S} \right )_{V,N}.$$
For more details, see Wikipedia:

http://en.wikipedia.org/wiki/Maxwell_relations

In thermodynamics, you only need a certain number of "independent" variables to fully define the state of the system. The other variables are then fixed by the equation of state. The number of independent variables, ##F## is given by the Gibbs phase rule:

##F=C-P+2##,

where ##C## is the number of chemical components in the system and ##P## is the number of phases in the system.

For example, the equation of state of an ideal gas is ##PV=nRT##, and let's suppose that it consists of a single chemical component. You only need to specify three of the four variables ##P,V,n,T## and the fourth is fixed by the equation of state. We can then denote partial derivatives of the variables with a notation like

##\left(\frac{\partial P}{\partial V}\right)_{n,T}=-\frac{nRT}{V^{2}}##,

where we specify that ##n## and ##T## are treated as constants in the differentiation. Here we fix enough variables to make sure that ##P## and ##V## are the only variables we don't know. You can easily imagine a process where we increase the volume of a closed ideal gas system while keeping temperature constant (a container with heat conducting walls), and the measure the corresponding change of pressure.

## 1. What are thermodynamic variables?

Thermodynamic variables are physical quantities that describe the state of a system at equilibrium. They include temperature, pressure, volume, and energy.

## 2. How are thermodynamic variables related to each other?

The relationships between thermodynamic variables are described by the laws of thermodynamics. For example, the first law states that energy cannot be created or destroyed, only transferred or converted from one form to another.

## 3. What is the difference between intensive and extensive thermodynamic variables?

Intensive variables, such as temperature and pressure, do not depend on the size or amount of the system. Extensive variables, such as volume and energy, do depend on the size or amount of the system.

## 4. How do thermodynamic variables affect the behavior of a system?

Changes in thermodynamic variables can cause a system to undergo phase changes, such as melting or vaporization, or to undergo chemical reactions. They also determine the direction and extent of heat and work transfer in a system.

## 5. What is the importance of understanding thermodynamic variables?

Understanding thermodynamic variables is crucial for predicting and controlling the behavior of physical systems, ranging from small molecules to large-scale industrial processes. It also provides a fundamental understanding of the laws and principles governing the behavior of matter and energy.

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