Specific heat at constant volume

AI Thread Summary
The discussion centers on the use of specific heat at constant volume, C_V, in deriving the equation for reversible adiabatic expansion, P_{1}V_{1}^{γ} = P_{2}V_{2}^{γ}. While C_V is defined for isovolumetric processes, it can still apply in adiabatic transformations because the change in internal energy, ΔU, depends solely on temperature for ideal gases. Participants clarify that during adiabatic processes, where heat transfer (dQ) is zero, the relationship dU = C_V dT holds true. This indicates that even though the process is not at constant volume, C_V can still be relevant due to the dependence of internal energy on temperature alone. The conversation highlights the nuances of thermodynamic principles in different types of processes.
Bipolarity
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C_{V} = \frac{∂U}{∂T}

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:

P_{1}V_{1}^{γ} = P_{2}V_{2}^{γ}

Why are we allowed to use C_{V} 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
\Delta U = nC_v\Delta T
The amount of heat is dependent on the type of process. It is Q = nC_v\Delta T
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
\Delta U = nC_v\Delta T
The amount of heat is dependent on the type of process. It is Q = nC_v\Delta T
only for constant volume process.

Superb! Thanks!

BiP
 
Bipolarity said:
C_{V} = \frac{∂U}{∂T}

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:

P_{1}V_{1}^{γ} = P_{2}V_{2}^{γ}

Why are we allowed to use C_{V} 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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