Question regarding using the expression $dE_{int}$=nC_vdT$

In summary: The first law of thermodynamics states that the change in Internal energy is equal to the sum of Heat gained or lost by the system and work done by the system or on the system. In an adiabatic process, Q=0, so $dE=-W$.
  • #1
Harikesh_33
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The first law of Thermodynamics states that the change in Internal energy is equal to the sum of Heat gained or lost by the system and work done by the system or on the system .

$dE=Q-W$...(1).

In an Adiabatic process ,Q=0 .

Therefore $dE=-W$ .

Now (https://phys.libretexts.org/Bookshelves/University_Physics/Book:_University_Physics_(OpenStax)/Book:_University_Physics_II_-_Thermodynamics_Electricity_and_Magnetism_(OpenStax)/03:_The_First_Law_of_Thermodynamics/3.07:_Adiabatic_Processes_for_an_Ideal_Gas)

here specific heat capacity at Constant volume is used instead of internal energy (ie) $dE=nC_vdT$ .

How can this specific heat be used here isn't the Volume changing (through Work done ?) .

How can $nC_vdT$=-dW$ be used ?
 
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  • #3
@Harikesh_33, it's worth noting that to make Latex render correctly here, you enclose the code between a pair of double hash-signs (##\text {##Your Latex code here##}##).

For example, doing this for nC_vT=-dW gives ##nC_vT=-dW##.

Or you can similarly use a pair of double dollar-signs to render the code on its own line. For example $$nC_vT=-dW$$Use the preview-toggle (top right on edit-toolbar) to check before posting.
 
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  • #4
In thermodynamics, the correct equation to use for ##C_v## is in terms of the internal energy E rather than heat Q: $$C_v=\frac{1}{n}\left(\frac{\partial E}{\partial T}\right)_V$$In the case of an ideal gas, E is a function only of T, and not V. So, for an ideal gas, we can write that $$dE=nC_vdT\tag{ideal gas}$$So, for an adiabatic reversible process of an ideal gas, we have $$dE=nC_vdT=-dW$$It's as simple as that.
 
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Related to Question regarding using the expression $dE_{int}$=nC_vdT$

1. What does the expression $dE_{int}$=nC_vdT represent?

The expression $dE_{int}$=nC_vdT represents the change in internal energy of a system, where n is the number of moles of gas, C_v is the molar specific heat at constant volume, and dT is the change in temperature.

2. How is the expression $dE_{int}$=nC_vdT used in thermodynamics?

The expression $dE_{int}$=nC_vdT is used to calculate the change in internal energy of a system during a process, such as heating or cooling. It is also used to determine the heat capacity of a substance.

3. What is the significance of the subscript v in the expression $dE_{int}$=nC_vdT?

The subscript v in the expression $dE_{int}$=nC_vdT represents the fact that the specific heat is measured at constant volume. This means that the volume of the system remains constant during the process.

4. How does the value of C_v differ from the value of C_p?

C_v and C_p represent two different types of heat capacity: molar specific heat at constant volume and molar specific heat at constant pressure, respectively. The value of C_p is always greater than C_v because it takes into account the work done by the system as it expands against external pressure.

5. Can the expression $dE_{int}$=nC_vdT be used for all substances?

No, the expression $dE_{int}$=nC_vdT is specifically for ideal gases. For other substances, the expression may need to be modified to take into account factors such as phase changes or non-ideal behavior.

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