Partial derivatives of thermodynamic state functions

In summary, a partial derivative is a mathematical concept used to measure the rate of change of a function with respect to one of its variables, while holding all other variables constant. Thermodynamic state functions, such as temperature, pressure, and internal energy, are measurable quantities that describe the state of a thermodynamic system. Partial derivatives are used in thermodynamics to analyze the behavior of these state functions and their relationships with each other. One example of a partial derivative in thermodynamics is the heat capacity at constant volume, which describes how the internal energy of a system changes as temperature changes. Partial derivatives are important in thermodynamics because they allow for quantitative study of thermodynamic systems and provide a framework for understanding their relationships.
  • #1
physlosopher
30
4
I'm in a first-year grad course on statistical mechanics and something about multivariable functions that has confused me since undergrad keeps popping up, mostly in the context of thermodynamics. Any insight would be much appreciated!

This is a general question, but as an example imagine you're given an equation of state of a thermodynamic system, specifically its internal energy in terms of temperature, volume, and number of particles (say there's one species of particle).
$$E=E(T,V,N)$$
My ultimate questions are, is the partial derivative ##(\frac {\partial E} {\partial V})_{T,N}## something equilibrium thermodynamics can tell us about, and is it (generally) the case that ##(\frac {\partial E} {\partial V})_{T,N}=-p##?

My immediate inclination is that if I want to say anything about pressure for this system, I should write down the exact differential of the internal energy using the First Law: ##dE = TdS - pdV + \mu dN##. Please correct me if I'm wrong: this exact differential suggests that the internal energy is a natural function of the entropy, volume, and number, and so if I want to relate ##-p## to the energy by a partial derivative, i.e. ##\frac {\partial E} {\partial V}=-p##, I really need to be holding the other natural variables constant - that is, I'm really interested in the partial ##(\frac {\partial E} {\partial V})_{S,N}=-p##, where the subscripts denote the variables held constant.

But I have E as a function of T, so if I want the pressure given my equation of state, I'd want to first write temperature in terms of entropy in order to express E in its natural variables, or apply a Legendre transform from the internal energy to the Helmholtz free energy (call it A), in which case I'd have the relationship ##(\frac {\partial A} {\partial V})_{T,N}=-p##, correct?

Thanks in advance for any help!
 
Science news on Phys.org
  • #2
As you say
[tex]A=E-TS[/tex]
[tex]dA=-SdT-pdV+\mu dN[/tex]
[tex](\frac{\partial A}{\partial V})_{T,N}=-p[/tex]
From here you can derive
[tex](\frac{\partial E}{\partial V})_{T,N}=-p+T(\frac{\partial S}{\partial V})_{T,N}[/tex]
 
  • Like
Likes physlosopher

1. What is a partial derivative in thermodynamics?

A partial derivative in thermodynamics is a measure of how a thermodynamic state function changes with respect to a specific independent variable, while holding all other variables constant. It is denoted by the symbol ∂ and is used to calculate the rate of change of a state function in a specific direction.

2. How is a partial derivative calculated?

A partial derivative is calculated by holding all other variables constant and taking the derivative of the state function with respect to the specific variable in question. This can be done using standard calculus techniques such as the chain rule and product rule.

3. What is the significance of partial derivatives in thermodynamics?

Partial derivatives are important in thermodynamics because they allow us to analyze how a thermodynamic state function changes in response to changes in specific variables. This can help us understand the behavior of a system and make predictions about its properties.

4. Can partial derivatives be negative in thermodynamics?

Yes, partial derivatives can be negative in thermodynamics. This indicates that the state function is decreasing in response to an increase in the specific variable. It is important to note that the sign of a partial derivative depends on the chosen direction of change and may differ for different variables.

5. How are partial derivatives used in thermodynamic equations?

Partial derivatives are used in thermodynamic equations to express the relationship between different state functions and variables. They allow us to manipulate these equations and solve for unknown variables, making it easier to analyze and understand thermodynamic systems.

Similar threads

I Change of Variables in Thermodynamics
    • Thermodynamics
Replies
4
Views
1K
B Question about the definition of a partial derivative
    • Thermodynamics
Replies
13
Views
786
I How to see this form for the chemical potential of an ideal gas?
    • Thermodynamics
Replies
6
Views
950
I Graphical difference between adiabatic and isothermal processes.
    • Thermodynamics
Replies
1
Views
651
I Does Callen's Entropy Expression for a General Ideal Gas Contain an Error?
    • Thermodynamics
Replies
19
Views
1K
I Obtaining this form for molar energy under virial expansion (Callen)
    • Thermodynamics
Replies
23
Views
1K
I Entropy and Heat Capacity Relation
    • Thermodynamics
Replies
2
Views
783
Thermodynamic equation of differentials (and how to work with them)
    • Thermodynamics
Replies
1
Views
764
I Maxwell’s Relations and Differentiating With Respect to ##N_i##
    • Thermodynamics
Replies
2
Views
465
Thermodynamics: Partial derivatives
    • Thermodynamics
Replies
4
Views
4K

Hot Threads

Recent Insights

Back
Top