Are Thermodynamic Equations Considered PDEs?

[MexChemE]

Hello, PF! As I was reading my P-Chem textbook, I noticed most thermodynamic equations involve partial derivatives, like these ones: $$C_V = {\left( \frac {\partial E}{\partial T} \right )}_V$$ $${\left( \frac {\partial H}{\partial T} \right )}_P = {\left( \frac {\partial E}{\partial T} \right )}_P + P{\left( \frac {\partial V}{\partial T} \right )}_P$$ However, none of these equations is ever actually called a PDE by the author. Is it implied they are PDEs given they involve partial derivatives, or are they not classified as PDEs such as the wave or heat equations? Thanks in advance!

 

[dextercioby]

They are not equations, but equalities, because there's no unknown function and no boundary conditions. In the first equality you wrote, you should determine what E(T,V) looks like, then partially differentiate wrt T, to get the function C_V(T,V). For the second, there's an equality involving 3 different functions, H(T,P), E(T,P) and V(T,P).

 

[MexChemE]

I get it now, they are not equations in the sense that they need not be solved, right? They are just showing the relation between thermodynamic functions.

 

[Mark44]

I get it now, they are not equations in the sense that they need not be solved, right? They are just showing the relation between thermodynamic functions.

Right.

 

[blue_raver22]

Hello there! It is great to see that you are exploring the use of partial differential equations (PDEs) in thermodynamics. PDEs are indeed very common in this field, as they allow us to describe how a system changes over multiple variables, such as temperature and pressure, at the same time.

To answer your question, it is important to understand the difference between an ordinary differential equation (ODE) and a PDE. ODEs involve derivatives with respect to a single variable, while PDEs involve derivatives with respect to multiple variables. In the equations you have provided, the derivatives are taken with respect to both temperature and pressure, making them PDEs.

However, it is not necessary for the author to explicitly label these equations as PDEs, as it is implied by the use of partial derivatives. The wave and heat equations, on the other hand, are specific types of PDEs that describe the behavior of waves and heat in a system, respectively.

I hope this clarifies your question. Keep exploring the use of PDEs in thermodynamics, as they are a powerful tool in understanding complex systems. Best of luck in your studies!