Specific heat at constant volume

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Discussion Overview

The discussion revolves around the application of specific heat at constant volume (CV) in the context of adiabatic processes, particularly regarding its use in deriving the equation for reversible adiabatic expansion of ideal gases. Participants explore the conditions under which CV can be applied and the implications for internal energy changes.

Discussion Character

  • Technical explanation
  • Debate/contested

Main Points Raised

  • One participant states that CV is defined as CV = ∂U/∂T and assumes it can only be used in constant volume processes, questioning its application in deriving the adiabatic expansion equation.
  • Another participant asks how CV is used to derive the equation for adiabatic transformation, seeking clarification and examples.
  • A participant reiterates the expression for change in internal energy for an ideal gas, stating that ΔU = nCVΔT, and notes that the heat transfer Q is only valid for constant volume processes.
  • Further clarification is provided that for an ideal gas, internal energy is a function of temperature, leading to the relation dU = CVdT, and in adiabatic expansion, dQ = 0, resulting in dU = CVdT = -pdV.

Areas of Agreement / Disagreement

Participants express uncertainty about the application of CV in adiabatic processes, with some agreeing on its definition and others questioning its use in this context. The discussion remains unresolved regarding the appropriateness of using CV for deriving equations related to adiabatic transformations.

Contextual Notes

There are limitations regarding the assumptions made about the applicability of CV in different processes, as well as the dependence on the ideal gas model. The discussion does not resolve the mathematical steps involved in the derivation of the adiabatic equation.

Bipolarity
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[tex]C_{V} = \frac{∂U}{∂T}[/tex]

This is the specific heat at constant volume so I assume it can only be used at constant volume. However, my textbook uses this to derive the following equation for reversible adiabatic expansion:

[tex]P_{1}V_{1}^{γ} = P_{2}V_{2}^{γ}[/tex]

Why are we allowed to use [itex]C_{V}[/itex] when it only works in isovolumetric processes?

BiP
 
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How is Cv used to derive the equation for adiabatic transformation?
Can you show it here?
 
The change in internal energy has the same expression for any process between two states. For ideal gas is
[tex]\Delta U = nC_v\Delta T[/tex]
The amount of heat is dependent on the type of process. It is [tex]Q = nC_v\Delta T[/tex]
only for constant volume process.
 
nasu said:
The change in internal energy has the same expression for any process between two states. For ideal gas is
[tex]\Delta U = nC_v\Delta T[/tex]
The amount of heat is dependent on the type of process. It is [tex]Q = nC_v\Delta T[/tex]
only for constant volume process.

Superb! Thanks!

BiP
 
Bipolarity said:
[tex]C_{V} = \frac{∂U}{∂T}[/tex]

This is the specific heat at constant volume so I assume it can only be used at constant volume. However, my textbook uses this to derive the following equation for reversible adiabatic expansion:

[tex]P_{1}V_{1}^{γ} = P_{2}V_{2}^{γ}[/tex]

Why are we allowed to use [itex]C_{V}[/itex] when it only works in isovolumetric processes?

BiP

For an ideal gas, the internal energy is a function only of temperature, such that dU = CvdT always. For an adiabatic expansion, dQ = 0, so that

dU = CvdT = -pdV
 

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