PV-work sign depends on the convention
The mistake
Mixing work done by the gas with work done on the gas in the same first-law equation.
Why it feels right
Both phrases use the word work, so the sign feels semantic rather than mathematical.
How to catch it
Expansion sometimes increases and sometimes decreases internal energy in your equations without a sign convention change.
The first principle
State your convention. With ΔU = Q − W_by, expansion gives W_by > 0. With the work-on-gas convention, use ΔU = Q + W_on.
worked mini-example
An insulated gas expands and does 20 J of work. What is ΔU using ΔU = Q − W_by?
Common wrong move
+20 J.
Correct
−20 J because Q = 0 and W_by = +20 J.
Test the repair TH-04 · transfer
include J
Make a fresh prediction before referring to the worked example.