Specific heat at constant volume

[Bipolarity]

$$C_{V} = \frac{∂U}{∂T}$$

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

 

[nasu]

How is Cv used to derive the equation for adiabatic transformation?
Can you show it here?

 

[Bipolarity]

How is Cv used to derive the equation for adiabatic transformation?
Can you show it here?

http://en.wikipedia.org/wiki/Adiaba...ous_formula_for_adiabatic_heating_and_cooling

BiP

 

[nasu]

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.

 

[Bipolarity]

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

 

[Chestermiller]

$$C_{V} = \frac{∂U}{∂T}$$

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 = C_vdT always. For an adiabatic expansion, dQ = 0, so that

dU = C_vdT = -pdV