# Thermodynamic process

## Main Question or Discussion Point

is it possible to have a process that is both adiabatic and isothermal?
i would appreciate if someone could explained that to me.
if possible, could you please give me an example in real life.

Thanks

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Andrew Mason
Homework Helper
is it possible to have a process that is both adiabatic and isothermal?
i would appreciate if someone could explained that to me.
if possible, could you please give me an example in real life.
If the "process" involves change, there can be no process that is both adiabatic and isothermal. You can show this mathematically from the first law and the ideal gas law:

dQ = dU + PdV (first law)

PV = nRT so d(PV) = nRdT (ideal gas law)

d(PV) = PdV + VdP = nRdT

so the first law becomes:

dQ = dU + nRdT - VdP

Now if dQ = 0 and dT = 0 then dU = VdP

But since dU = nCvdT, and dT = 0 then dP must be 0.

Also, if dT = 0 and dQ = dU + PdV = nCvdT + PdV = 0, then dV = 0

So the only adiabatic and isothermal process would be one in which dT, dV and dP are all 0. There is no change at all.

AM

Q_Goest
Homework Helper
Gold Member
I'd agree with most of what Andrew wrote, that's a good primer for understanding the application of the first law here. Note that we're interested here in processes in which pressure will change. However, the caveat I'd add is that a process CAN be isothermal and adiabatic if the fluid is incompressible.

All fluids, be they liquids or gasses, are compressible to some minor degree. So all fluids will heat up according to the equations provided by Andrew above. However, the more incompressible they are, the less they will increase in temperature.

Consider the isentropic compression of water. Starting at atmospheric pressure and 70 F and compressing to 50 psig, will result in the pressure increasing by a factor of roughly 500%. The temperature increase on the other hand, is on the order of 0.0053 degrees F, an increase in absolute temperature of only 0.001%.

So to answer your question, an incompressible fluid can go through a process which is adiabatic and isothermal.

siddharth
Homework Helper
Gold Member
is it possible to have a process that is both adiabatic and isothermal?
i would appreciate if someone could explained that to me.
You need to be more specific. Is the system open or closed? Does it involve an ideal gas? If it's an ideal gas, what AM posted holds.

If the "process" involves change, there can be no process that is both adiabatic and isothermal. You can show this mathematically from the first law and the ideal gas law:

dQ = dU + PdV (first law)

PV = nRT so d(PV) = nRdT (ideal gas law)

....
In a more general case, for a single component, single phase closed system, the internal energy will be a function of 2 variables. (say volume and temperature).

From the first law,
$$\delta U = \delta Q + \delta W$$

The infinitesimal change in the Internal Energy dU for a general process will be,
$$dU = \left(\frac{\partial U}{\partial V\right)_T dV + \left(\frac{\partial U}{\partial T}\right)_V dT$$

For an adiabatic process, by definition, $$\delta Q = 0$$.
For an isothermal process, dT=0.

So it's possible to have a process which is both isothermal and adiabatic, and where the internal energy change is non-zero.

Last edited:
Andrew Mason