Thermodynamics: Show that the two relations give Pv = RT

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Homework Help Overview

The discussion revolves around demonstrating that two given relations for an ideal gas lead to the equation Pv = RT. The subject area is thermodynamics, specifically focusing on the behavior of gases under constant temperature and volume conditions.

Discussion Character

  • Exploratory, Mathematical reasoning

Approaches and Questions Raised

  • Participants explore the use of partial derivatives and their implications for the equations of state for gases. There is an attempt to manipulate the equations by equating and adding them, though uncertainty about the process is expressed. One participant suggests treating the equations as ordinary differential equations and finding functions of the constants involved.

Discussion Status

The discussion is ongoing, with participants sharing their attempts and questioning their understanding of the mathematical operations required. Some guidance has been offered regarding integration and manipulation of the equations, but no consensus or resolution has been reached yet.

Contextual Notes

One participant notes a lack of familiarity with partial derivatives, which may be impacting their ability to progress. The constraints of the homework context are acknowledged, as participants navigate the rules of the problem without providing direct solutions.

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Homework Statement



For an ideal gas the slope of an isotherm is given by

(∂P/∂v) constant T = -P/v

and that of an isochore is

(∂P/∂T) constant v = P/T

Show that these relations give Pv = RT

Homework Equations



Pv = RT

The Attempt at a Solution



I have never worked with partial derivatives before encountering this problem so I am unfamiliar with the rules and operations involved. I tried setting them equal, adding them to each other but I just don't know where I am going.
 
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You can treat each of the equations as an ordinary differential equation where the independent variable is V and T respectively. When you solve them, you will have two constants. But these constants must be then functions of the "constant" variable, T and V respectively. Then you should be able to find those functions and get the ideal gas law.
 
Ok so I got up to this step.

dP = -P/v dv + P/T dt

Im unsure of how to proceed from here
 
Divide by P and then integrate.
 
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