Using Constant-Volume Specific Heat for Calculating Change of Internal Energy?

In summary, it is always true to use constant-volume specific heat to calculate the change of internal energy for an ideal gas, regardless of the process. This is due to the unique non-interaction of atoms in an ideal gas. The definition of heat capacity is the heat needed to increase the temperature of a system by one degree under a specific condition, and this can be represented by both the molar and specific heat capacity.
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KFC
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I note that in many case, we always use the constant-volume specific heat to calculate the change of internal energy.

For example, in a adiabatic process (P1, V1, T1) to (P2, V2, T2), since internal energy is state variable, we always like to build a fictitious isochoric process from (P1, V1, T1) to (P1', V1, T2) and isobaric process from (P1', V1, T2) to (P2, V2, T2) so that the total change of internal energy be

[tex]\Delta U = n C_v(T2-T1) + nC_p(T2-T2) = n C_v(T2-T1)[/tex]

Is this always true? There is a chapter about isobaric process in my text. The author use the constant-volume specific heat to calculate the change of internal energy

[tex]\Delta U = nC_v \Delta T[/tex]

the work done by the ideal gas is

[tex]\Delta W = nR\Delta T[/tex]

according to first law

[tex]\Delta Q = \Delta U + \Delta W = nC_v\Delta T + nR\Delta T = nC_p \Delta T[/tex]

this results is really confusing me. I wonder why don't we just use the constant-pressure specific heat to calculate the change of internal energy for isobaric process? But if we use [tex]C_p[/tex] to calculate [tex]\Delta U[/tex], the result will be different ... well all of these doubts is concluded in the following questions:

1) will it ALWAYS be true to use constant-volume specific heat to calculate the change of internal energy? No matter what process is concerned (even for isobaric process)?

2) The definition of heat capacity is: the change of heat per mole per degree. So why we keep use specific heat to calculate the change of internal energy instead of the change heat?


Thanks.
 
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KFC said:
1) will it ALWAYS be true to use constant-volume specific heat to calculate the change of internal energy? No matter what process is concerned (even for isobaric process)?

Only for an ideal gas, https://www.physicsforums.com/showpost.php?p=2009417&postcount=2". This is a result of the unique non-interaction of atoms in an ideal gas.

It might help to look at it this way: It's always true, for any system, that

[tex]dU=T\,dS-d(PV)+V\,dP[/tex]

For a constant-pressure process ([itex]dP=0[/itex]), the heat capacity

[tex]C_P=T\left(\frac{\partial S}{\partial T}\right)_P=C_V+nR[/tex]

for an ideal gas, and [itex]d(PV)=nR\,dT[/itex] for an ideal gas. So

[tex]\left(\frac{\partial U}{\partial T}\right)_P=C_V[/tex]

As you can see, [itex]dU=C_V dT[/itex] holds for constant-volume and constant-pressure processes. In fact it holds for all processes (for an ideal gas).

KFC said:
2) The definition of heat capacity is: the change of heat per mole per degree. So why we keep use specific heat to calculate the change of internal energy instead of the change heat?

The definition of heat capacity is [itex]T\left(\frac{\partial S}{\partial T}\right)_X[/itex], the heat needed to increase the temperature of a system by one degree under some condition X (there are also the molar heat capacity and specific heat capacity, which are normalized by amount of matter and mass, respectively).
 
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  • #3
Mapes said:
Only for an ideal gas, https://www.physicsforums.com/showpost.php?p=2009417&postcount=2". This is a result of the unique non-interaction of atoms in an ideal gas.

It might help to look at it this way: It's always true, for any system, that

[tex]dU=T\,dS-d(PV)+V\,dP[/tex]

For a constant-pressure process ([itex]dP=0[/itex]), the heat capacity

[tex]C_P=T\left(\frac{\partial S}{\partial T}\right)_P=C_V+nR[/tex]

for an ideal gas, and [itex]d(PV)=nR\,dT[/itex] for an ideal gas. So

[tex]\left(\frac{\partial U}{\partial T}\right)_P=C_V[/tex]

As you can see, [itex]dU=C_V dT[/itex] holds for constant-volume and constant-pressure processes. In fact it holds for all processes (for an ideal gas).



The definition of heat capacity is [itex]T\left(\frac{\partial S}{\partial T}\right)_X[/itex], the heat needed to increase the temperature of a system by one degree under some condition X (there are also the molar heat capacity and specific heat capacity, which are normalized by amount of matter and mass, respectively).

Thanks again. It is really helpful.
 
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Related to Using Constant-Volume Specific Heat for Calculating Change of Internal Energy?

1. What is internal energy?

Internal energy is the total amount of energy stored within a system. It includes the kinetic energy of particles, potential energy of bonds between particles, and thermal energy.

2. How does internal energy change?

Internal energy can change through various processes, such as heating or cooling, chemical reactions, or mechanical work.

3. What is the formula for calculating change in internal energy?

The formula for change in internal energy (ΔU) is ΔU = Q + W, where Q is the heat added to the system and W is the work done on the system.

4. How is change in internal energy related to temperature?

According to the first law of thermodynamics, the change in internal energy is equal to the heat added to the system minus the work done by the system. This means that an increase in internal energy will result in an increase in temperature, and vice versa.

5. Can internal energy be negative?

Yes, internal energy can be negative. This means that the system has lost energy, either through heat transfer out of the system or work done by the system. However, the change in internal energy (ΔU) can never be negative, as it is a difference between two values and will always be a positive or zero value.

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