Heat, work, and internal energy

Heat and work are transfers, not stored substances

Internal energy U belongs to the state; heat Q and work W describe ways energy crosses the boundary. The first law is bookkeeping that becomes predictive only after the sign convention and system are specified.

Derive the first-law balance

For a closed system using W as work done by the system, energy conservation gives ΔU = Q − W. Heat entering raises U; work done by the system lowers U unless compensated by heat. For an ideal gas, U depends only on T, so ΔU = nC_VΔT regardless of the path.

A gas receives Q = 500 J and does W = 180 J of expansion work. ΔU = 500 − 180 = 320 J. If nC_V = 40 J/K, its temperature rises by 8.0 K.

Sign errors are boundary errors

If W is defined as work done on the system, the equation becomes ΔU = Q + W_on. Both forms are correct; mixing them mid-calculation is not. A cyclic process has ΔU = 0 even though Q and W can each be nonzero.

Work it: A gas is compressed by 250 J while 100 J of heat leaves. Using W_by = −250 J and Q = −100 J, calculate ΔU and explain the temperature trend for an ideal gas.