Adiabatic Processes: Why does this equation use constant volume and not constant pressure?

[abdossamad2003]

Why is this equation (red sign) written in constant volume and not in constant pressure?

 

[Hill]

Because this is what counts for the internal energy of the gas.

 

[abdossamad2003]

I was not convinced why internal energy should be written for heat capacity in constant volume. This process does not take place in constant volume, and if it is in constant volume, the change in internal energy must be zero.

 

[abdossamad2003]

The work is done on the adiabatic system, that is, it is not in constant volume, in my opinion, constant pressure seems more reasonable

 

[Chestermiller]

What is your understanding of the effect of volume on the internal energy of an ideal gas?

 

[abdossamad2003]

The effect of volume on the internal energy is meaningful only in diabatic processes, for example, when heat is added to the system at a constant volume and the internal energy increases, but in adiabatic processes, when the volume is constant, the work done on the system is zero and the incoming heat is zero, as a result of the change in energy Internal is zero.

 

[Chestermiller]

The effect of volume on the internal energy is meaningful only in diabatic processes, for example, when heat is added to the system at a constant volume and the internal energy increases, but in adiabatic processes, when the volume is constant, the work done on the system is zero and the incoming heat is zero, as a result of the change in energy Internal is zero.

This is totally incorrect. Irrespective of the process, the internal energy of an ideal gas depends only on temperature, and not volume.

 

[Hill]

I was not convinced why internal energy should be written for heat capacity in constant volume. This process does not take place in constant volume, and if it is in constant volume, the change in internal energy must be zero.

See, e.g., here: https://en.wikipedia.org/wiki/Internal_energy#Internal_energy_of_the_ideal_gas. $C_V$ is the coefficient of proportionality between internal energy on one hand and number of moles and temperature of the gas on the other hand.