# Doesn't $dP×dV$ contribute to internal energy change?

When the heat supplied to system is zero and change in internal energy is brought by work, it is said that $$W=P(ext)×dV$$ is the work done.

Isn't that work done (causing internal energy change) due to change in pressure $$P_{ext}-P_{int}$$?

Isn't it $$W=dP×dV$$?

• There were a few comments here 8 days ago. Why were they deleted? Commented Oct 4, 2020 at 7:28
• I think you should change W into dW and I guess dP is equal to d(Pext--Pint). Commented Oct 10, 2020 at 4:25
• P(int) varies if you push (or pull) the piston of the container. P(ext) (in your first equation this isn't an infinitesimal difference) can stay constant: dW=P(ext) x dV. The dP in dW=dP x dV is indeed the (infinitesimal) difference d(Pext - Pint) Commented Oct 10, 2020 at 9:33

Let us consider a system of gas in a container with a massless , frictionless and movable piston which is in equilibrium with the surrounding.

Now , From first law of thermodynamics , we know that

$$dQ = dU + dW$$.

$$Q$$ here indicates the energy flow due to temperature difference between the system and the surrounding.

But since you assumed $$Q$$ to be zero this means that the system is in thermal equilibrium with the surrounding.

Now if you want to do some work with minimal loss of energy due to friction, then we proceed with a quasi static process or say a reversible process in which the system is in equilibrium with the surrounding all the time.

Now initially the system was in equilibrium , so

$$P_{ext} = P_{gas} = P$$

Now if you bring an infinitesimal change in the external pressure , say $$dP$$, i.e the external pressure becomes $$P+dP$$ then the volume of the gas changes by an amount of $$dV$$. But since it was an infinitesimal change , the gaseous pressure quickly equalises the external pressure.

So, work done by you is

$$dW = (P+dP) . dV = P.dV + dP.dV$$

The term $$dP.dV$$ is a very... small number and so we generally ignore it.

The main thing to note here is that the energy which you gave was because of the external pressure $$(P + dP)$$ and not $$dP$$. So you must include $$(P + dP)$$ term in the equation for work and not $$dP$$ only.

One more thing to note here is that the energy given will change the internal energy and thus the temperature will change but assuming isothermal process this exact amount of energy is released and the system is again in thermal equilibrium with the surrounding.

Since the system is again in equilibrium , the internal pressure equals the external pressure i.e.

$$P_{gas} = P_{ext} + dP = P + dP$$

Now if you decrease the external pressure by an amount of $$dP$$ , the gas will again reach its initial condition by doing work on the surrounding and thus it is reversed.

So, you can say that the work done on the system is given by

$$W_{ext} = \int (P + dP) × dV = \int P.dV$$

Isn't that work done (causing internal energy change) due to change in pressure

Yes it is . But that work was done by the external pressure and not by that changed amount alone. And since you are calculating the work done on the gas you should include the external pressure only and not the internal .

Hope it helps 🙂.

• I have a question. The change in kinetic energy of the block would be 6J right? So why isn't dU equal to the net work? Commented Oct 4, 2020 at 7:59
• But if you ask for the net work, it will be 6J because friction opposed your force. The net work will still be 10 J as heat is developed (as well as kinetic energy of the block) due to friction. Commented Oct 10, 2020 at 3:12
• Well, why isn't the net work 10J? You push the block with 5N over 2m. So the block is accelerated with a force of 3N, thereby acquiring kinetic energy. But you push with 5N, which means also heat is developed. Why substracting this heat? In the question it is said no heat is supplied. Do you think friction is present when compressing the gas? ;) Commented Oct 10, 2020 at 3:31
• But will this friction energy (heat) not flow (at least a part of it) into the system? Commented Oct 10, 2020 at 3:40
• You write: also the case in your question is isothermal since you are not allowing heat transfer. But isn't internal energy added by the work done? If I pump my bike's tire, the metal gets very hot because the gas is heated. Commented Oct 10, 2020 at 3:49