Deriving the TdS Equation for Thermal Physics

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SUMMARY

The discussion focuses on deriving the TdS equation in thermal physics, specifically the equation TdS = C_{V} (∂T/∂P)_{V}dP + C_{P} (∂T/∂V)_{P}dV. Key concepts include the relationships between internal energy (dU = δQ - δW) and heat capacities (C_{P} and C_{V}). The participants emphasize the importance of treating entropy (S) as a function of pressure (P) and volume (V), leading to the partial derivatives of S with respect to P and V. The derivation hinges on manipulating these relationships to express TdS accurately.

PREREQUISITES
  • Understanding of thermodynamic equations and concepts
  • Familiarity with the first law of thermodynamics
  • Knowledge of heat capacities, specifically C_{P} and C_{V}
  • Proficiency in partial derivatives and their applications in thermodynamics
NEXT STEPS
  • Study the derivation of the first law of thermodynamics in detail
  • Explore the relationship between entropy and temperature in closed systems
  • Learn about Maxwell's relations and their applications in thermodynamics
  • Investigate the implications of the TdS equation in real-world thermodynamic processes
USEFUL FOR

Students and professionals in physics, particularly those focusing on thermal physics, thermodynamics, and related engineering fields. This discussion is beneficial for anyone looking to deepen their understanding of entropy and its role in thermodynamic equations.

Elzair
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Homework Statement


Derive the following equation


Homework Equations


TdS = C_{V} \left( \frac{\partial T}{\partial P} \right)_{V}dP + C_{P} \left( \frac{\partial T}{\partial V} \right)_{P}dV


The Attempt at a Solution



dU = \delta Q - \delta W

\delta Q = TdS for a closed system

C_{P} = T \left( \frac{\partial S}{\partial T} \right)_{P}

C_{V} = T \left( \frac{\partial S}{\partial T} \right)_{V}

I am not sure where to go from here.
 
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I think the trick here is playing around with the relations.

If you consider S as a function of p and v as independent variables, then
dS = \left(\frac{\partial S}{\partial P}\right)_V dP + \left(\frac{\partial S}{\partial V}\right)_P dV

But, \frac{\partial S}{\partial P}_V = \left(\frac{\partial S}{\partial T}\right)_V \left(\frac{\partial T}{\partial P}\right)_V, and so on.
 

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