Partial differentiation - Constants

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SUMMARY

The discussion centers on the application of partial differentiation in thermodynamics, specifically the equality (\frac{∂S}{∂V})T = (\frac{∂P}{∂T})V. The user seeks clarification on which variables remain constant when rearranging the equation to (\frac{∂S}{∂P})? = (\frac{∂V}{∂T})?. It is established that when differentiating with respect to one variable, all other variables should be treated as constants, a fundamental principle in thermodynamic analysis. The discussion also references the cyclic relation in partial derivatives, emphasizing its importance in understanding these relationships.

PREREQUISITES
  • Understanding of partial differentiation in thermodynamics
  • Familiarity with the concepts of entropy (S), volume (V), pressure (P), and temperature (T)
  • Knowledge of the cyclic relation in partial derivatives
  • Basic grasp of thermodynamic equations and their applications
NEXT STEPS
  • Study the implications of the cyclic relation in partial derivatives
  • Learn about the Maxwell relations in thermodynamics
  • Explore the derivation and applications of the triple product rule
  • Investigate the role of state functions in thermodynamic systems
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Students and professionals in thermodynamics, particularly those studying or working in fields related to physical chemistry, engineering, or any discipline requiring a solid understanding of thermodynamic principles and partial differentiation.

Jalo
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Homework Statement



Consider the following equality:

([itex]\frac{∂S}{∂V}[/itex])T = ([itex]\frac{∂P}{∂T}[/itex])V

If I rearrange the equality so that I write:

([itex]\frac{∂S}{∂P}[/itex])? = ([itex]\frac{∂V}{∂T}[/itex])?

What variables will be constant in each side?
I'm having some trouble in a few thermodynamics problems because of this...

Homework Equations





The Attempt at a Solution



I don't know how to do this. Normally I just ignore and assume that every variable except the variable of differentiation is a constant.
If anyone could confirm or correct me I'd be thankful.

Thanks.
 
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Jalo said:

Homework Statement



Consider the following equality:

([itex]\frac{∂S}{∂V}[/itex])T = ([itex]\frac{∂P}{∂T}[/itex])V

If I rearrange the equality so that I write:

([itex]\frac{∂S}{∂P}[/itex])? = ([itex]\frac{∂V}{∂T}[/itex])?

What variables will be constant in each side?
I'm having some trouble in a few thermodynamics problems because of this...

Homework Equations



The Attempt at a Solution



I don't know how to do this. Normally I just ignore and assume that every variable except the variable of differentiation is a constant.
If anyone could confirm or correct me I'd be thankful.

Thanks.
The following may help you.

For the case in which three variables can be related by a function of the form f(x, y, z) = 0, then the following relations hold.

[itex]\displaystyle \left(\frac{\partial x}{\partial y}\right)_z\left(\frac{\partial y}{\partial z}\right)_x\left(\frac{\partial z}{\partial x}\right)_y = -1[/itex]

[itex]\displaystyle \left(\frac{\partial x}{\partial y}\right)_z = \frac{1}{\left(\frac{\partial y}{\partial x}\right)_y}[/itex]

See the following two Wikipedia entries:

http://en.wikipedia.org/wiki/Triple_product_rule

http://en.wikipedia.org/wiki/Exact_differential#Cyclic_relation
 

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