According to the first law of thermodynamics, which expression correctly relates the change in internal energy to heat and work, using q for heat added to the system and w for work done on the system?

• A. ΔU = q + w
• B. ΔU = q - w
• C. ΔU = -q - w
• D. ΔU = w - q

Answer: (A)

The main idea here is how energy transfers are accounted for in the first law with a specific sign convention. The internal energy change ΔU equals the energy entering the system minus the energy leaving it, and with q defined as heat added to the system and w defined as work done on the system, both forms of energy transfer add to the system’s energy. So the correct relation is ΔU = q + w.

If you heat the system by a amount q, that energy goes into increasing internal energy. If you apply external work to compress the system, that also adds energy to the system, contributing w to ΔU. Conversely, if the system does work on its surroundings (which would make w negative under this convention), energy leaves the system and ΔU decreases accordingly. For example, q = +10 J and w = +5 J give ΔU = +15 J, while if the system does work on the surroundings by 5 J with no heat, w = -5 J and ΔU = -5 J.

This is why the expression with a plus sign between q and w matches the given definitions.