Evolution of temperature (adiabatic procsses)

[ted1986]

Hello,

I'm trying to derive a differential equation as requested in the attached exercise (thermal1.jpg).
I'm not quite sure my solution is the right answer (my_solution1.jpg).
How do I get rid of the energy dU ?

Thnks


Ted

 

[sgd37]

well in an adiabatic processes dQ = 0 so ds = 0 giving you quite a simple derivation of the differential equation

 

[ted1986]

well in an adiabatic processes dQ = 0 so ds = 0 giving you quite a simple derivation of the differential equation

OK, It sure gives a simple derivation of the differential equation, but what is the physical explanation for that?

Thanks

 

[sgd37]

it most generally means that it is a reversible process

 

[paranormal]

Hello Ted,

Thank you for your question. The evolution of temperature in adiabatic processes can be described using the first law of thermodynamics, which states that the change in internal energy (dU) of a system is equal to the heat (Q) added to the system minus the work (W) done by the system. In adiabatic processes, no heat is exchanged with the surroundings, so Q = 0. This means that the change in internal energy is solely due to the work done by the system, which can be calculated using the ideal gas law (PV = nRT). By substituting this into the first law of thermodynamics, we can derive the differential equation dT/dt = - (gamma-1)/Cv * T * dV/dt, where gamma is the heat capacity ratio and Cv is the specific heat at constant volume. This equation describes the change in temperature with respect to time in an adiabatic process.

I hope this helps and please let me know if you have any further questions.

Best,