Specific heat at constant volume

In summary: RT dV/V, orCv dT/T = -nR dV/V, which gives the adiabatic equation, P1V1^γ = P2V2^γ, with γ = Cp/Cv. This shows that Cv is used to derive the equation for adiabatic transformation, as demonstrated on Wikipedia. In summary, the specific heat at constant volume (Cv) is used to derive the equation for reversible adiabatic expansion, as it is a function of temperature and can be used to calculate the internal energy for an ideal gas in a constant volume process.
  • #1
Bipolarity
776
2
[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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  • #2
How is Cv used to derive the equation for adiabatic transformation?
Can you show it here?
 
  • #4
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.
 
  • #5
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
 
  • #6
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
 

Related to Specific heat at constant volume

1. What is specific heat at constant volume?

Specific heat at constant volume, also known as the molar heat capacity at constant volume, is the amount of heat required to raise the temperature of one mole of a substance by one degree Celsius while keeping the volume constant.

2. How is specific heat at constant volume measured?

Specific heat at constant volume is typically measured using a calorimeter, which is a device that measures the heat exchanged between the substance and its surroundings. The substance is placed in the calorimeter and heated, and the change in temperature is recorded. From this, the specific heat at constant volume can be calculated.

3. What is the difference between specific heat at constant volume and specific heat at constant pressure?

The main difference between specific heat at constant volume and specific heat at constant pressure is that at constant volume, the volume of the substance remains constant while the temperature changes, whereas at constant pressure, the pressure remains constant while the temperature changes. Additionally, specific heat at constant pressure tends to be higher than specific heat at constant volume.

4. Why is specific heat at constant volume important?

Specific heat at constant volume is an important property of substances because it helps us understand how much energy is required to change the temperature of a substance. It is also used in thermodynamic calculations, such as in the calculation of enthalpy and entropy changes.

5. How does specific heat at constant volume relate to the internal energy of a substance?

The specific heat at constant volume is related to the internal energy of a substance through the equation: ΔU = nCvΔT, where ΔU is the change in internal energy, n is the number of moles, Cv is the specific heat at constant volume, and ΔT is the change in temperature. This shows that the internal energy of a substance increases as its temperature increases, and the specific heat at constant volume determines how much energy is required to produce this change in temperature.

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