Derive the fundamental equation from and equation of state

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SUMMARY

The discussion focuses on deriving the fundamental equation from the Van der Waals equation of state for fluids, specifically given by P=\frac{RT}{v-b}-\frac{a}{v^2} and u=cRT-\frac{a}{v}. The fundamental equation is expressed as dS=\frac{1}{T}dU+\frac{P}{T}dv. The user successfully reformulates the internal energy equation but encounters difficulties integrating the resulting equation of two variables. It is clarified that the fundamental equation is independent of the equation of state and is derived from the second law of thermodynamics.

PREREQUISITES
  • Understanding of the Van der Waals equation of state
  • Familiarity with thermodynamic concepts such as internal energy and entropy
  • Knowledge of calculus, particularly integration of multivariable functions
  • Basic principles of the second law of thermodynamics
NEXT STEPS
  • Study the derivation of the fundamental equation from thermodynamic principles
  • Learn about integrating multivariable functions in calculus
  • Explore the implications of the second law of thermodynamics on thermodynamic equations
  • Investigate the properties and applications of Van der Waals fluids in thermodynamics
USEFUL FOR

Students and professionals in thermodynamics, physicists, and engineers working with fluid dynamics and state equations will benefit from this discussion.

babbagee
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Hello,

I just need help deriving the fundamental equation from Equation of state for wan der waals fluids.

P=\frac{RT}{v-b}-\frac{a}{v^2}

and

u=cRT-\frac{a}{v}

where a,b, and c are constants.

I know the fundamental equation is

dS=\frac{1}{T}dU+\frac{P}{T}dvI solved the u equation so it's in terms of u and v and just plugged in the P equation to \frac{P}{T} but then I get an equation of two variables which I am not sure how to integrate. So I get an equation in the form of

dS=f(u,v)dU + f(u,v)dv

Can someone help me with the integration please.

Thanks in advance!
 
Last edited:
Science news on Phys.org
The fundamental equation is independent of the equation of state, and follows from the more general 2nd law of thermodynamics. So it is not clear what is being asked here.
 

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