# Does doing work on system increase or decrease its internal energy?

I am currently reading about the 1st law of thermodynamics, from the book "A Hot Story" by G. Venkataraman. While discussing the law, the author tells that when we do work on the system, its internal energy increases. The internal energy also increases when we impart heat into the system. So that the 1st law reads: $$$$\tag{1} \Delta U= \Delta Q+ \Delta W$$$$

I have no problem understanding this part. However, he later discusses an example where we do work on gas at a constant pressure, the volume of the gas decreases. Correspondingly, the 1st law reads:

$$$$\tag{2} \Delta U= \Delta Q-P \Delta V$$$$

But I see a conflict between the interpretations of the equations $$1$$ and $$2$$. According to eqn $$1$$, doing work on the gas should increase the internal energy, whereas eqn $$2$$ suggests that work done on the gas decreases the internal energy. However it seems to me that by doing work on the system, I am imparting energy to the system, which should increase the internal energy.

Can anyone please suggest as to how should this confusion be resolved?

If $$\Delta V$$ is negative due to compression of the gas, then the sign of the work is positive and the internal energy will increase. The convention used here is that the sign of the work done on a system corresponds with the energy of the system, as you rightly interpreted. There is no conflict here at all.

Does doing work on system increase or decrease its internal energy?

Doing work on the system always transfers energy to the system, thus adding to the internal energy of the system. But whether or not the internal energy of the system increases, decreases, or remains the same depends not only on the work done on the system, but also on any energy transferred into or out of the system in the form of heat.

So that the 1st law reads: $$$$\Delta U= \Delta Q+ \Delta W$$$$

I don't know if this equation is yours or is from the referenced book, but it is incorrectly written. It should be written as

$$\Delta U=Q+W$$

$$\Delta$$ means "change in", which means a change in a system property, such as internal energy, pressure, and volume. Heat and work are not system properties. There is no "change" in heat or work. There are quantities of energy transferred to or from the system in the form of heat and work.

But I see a conflict between the interpretations of the equations $$1$$ and $$2$$. According to eqn $$1$$, doing work on the gas should increase the internal energy, whereas eqn $$2$$ suggests that work done on the gas decreases the internal energy.

Work done on the system is compression work. For compression work the volume decreases, i.e., $$\Delta V\lt 0$$ making it a negative quantity. Substituting a negative value for $$W$$ in the above equation makes the work done on the system positive, thus increasing internal energy. So there is no conflict.

Hope this helps.