Derive the fundamental equation from and equation of state

In summary, the conversation discusses deriving the fundamental equation from the equation of state for van der Waals fluids. The fundamental equation is independent and follows from the 2nd law of thermodynamics. The speaker is unsure how to integrate an equation with two variables and requests assistance.
  • #1
babbagee
74
0
Hello,

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

P=[itex]\frac{RT}{v-b}[/itex]-[itex]\frac{a}{v^2}[/itex]

and

u=cRT-[itex]\frac{a}{v}[/itex]

where a,b, and c are constants.

I know the fundamental equation is

dS=[itex]\frac{1}{T}[/itex]dU+[itex]\frac{P}{T}[/itex]dvI solved the u equation so it's in terms of u and v and just plugged in the P equation to [itex]\frac{P}{T}[/itex] 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:
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  • #2
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.
 

1. What is the fundamental equation in thermodynamics?

The fundamental equation in thermodynamics is a mathematical equation that describes the relationship between various thermodynamic properties, such as temperature, pressure, and volume, of a system in equilibrium. It is derived from the laws of thermodynamics and is considered the cornerstone of thermodynamics.

2. How is the fundamental equation derived from an equation of state?

The fundamental equation is derived from an equation of state by using the laws of thermodynamics and mathematical manipulations. An equation of state is a relationship between the macroscopic properties of a system, while the fundamental equation takes into account microscopic properties and their effects on the system. Therefore, the derivation involves incorporating additional terms and variables to account for these microscopic properties.

3. What are the different forms of the fundamental equation?

There are various forms of the fundamental equation, depending on the type of system and the properties being considered. Some common forms include the internal energy form, the entropy form, and the enthalpy form. Each form is derived by manipulating the fundamental equation to express a specific property as the dependent variable.

4. Can the fundamental equation be used for all types of systems?

Yes, the fundamental equation can be used for all types of systems, including closed, open, and isolated systems. However, the form of the equation may differ depending on the type of system and the properties being considered. In some cases, additional equations or constraints may need to be applied to accurately describe the system's behavior.

5. What is the significance of the fundamental equation in thermodynamics?

The fundamental equation is significant in thermodynamics because it provides a comprehensive and mathematically rigorous description of a system's behavior. It allows scientists to predict how a system will respond to changes in its environment and to calculate important thermodynamic quantities such as heat, work, and internal energy. It also serves as the basis for various thermodynamic principles and laws.

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