On the notion of pressure in the canonical ensemble

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SUMMARY

The discussion centers on the calculation of pressure in a canonical ensemble, specifically the relationship between pressure (P) and internal energy (U) with respect to volume (V). The formula P = dU/dV is derived from thermodynamic principles, where dU = dQ + dW. Although the volume is fixed in a canonical ensemble, the concept of pressure is defined through infinitesimal changes in volume, allowing for the calculation of pressure as a limit of the energy change per unit volume change.

PREREQUISITES
  • Understanding of canonical ensemble in statistical mechanics
  • Familiarity with thermodynamic principles, specifically dU = dQ + dW
  • Knowledge of calculus, particularly differentiation
  • Basic concepts of pressure in thermodynamics
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  • Study the implications of infinitesimal changes in thermodynamic systems
  • Explore the derivation of thermodynamic identities in statistical mechanics
  • Learn about the relationship between pressure and volume in different thermodynamic ensembles
  • Investigate the concept of undefined values in mathematical limits
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Students and professionals in physics, particularly those studying statistical mechanics and thermodynamics, as well as researchers interested in the mathematical foundations of pressure in canonical ensembles.

Derivator
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Hi folks,

since the volume V is fixed in a canonical ensemble I'm a bit confused about the fact, that the pressure is calculated as the derivation of the internal energy U with respect to the volume V.

Sure, P = dU/dV comes from dU = dQ + dW = tdS - pdV + ... But what does it mean to derivate with respect to the volume, when the volume of the system can't be changed, since it is fixed. (by the way: when the volume is fixed, pdV= p*0 =0, so p is "undefined", since it can take any value.)

--
derivator
 
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The volume is not free to change by any significant amount, but we are considering the case where it is changed by an infinitesimal amount. In response, the system energy changes by an infinitesimal amount. The ratio is the pressure.
 

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