The Closed-System Energy Balance, Step by Step

A closed system exchanges energy but not mass. The first law for a closed system is one equation, and almost every piston-cylinder problem is just that equation plus a process constraint.

The equation

Q − W = m·(u2 − u1)

Sign convention (the one this site and most US textbooks use):

• Q > 0 — heat into the system.
• W > 0 — work done by the system.

So if you add heat and the system does no work, all of it goes into internal energy.

The process constraint

The energy balance has two unknowns (Q and W) once you know the states. The process supplies the second relation:

Process	Constraint	Boundary work Wb
Constant volume	v2 = v1	0
Constant pressure	P2 = P1	P·(V2 − V1)
Isothermal (ideal gas)	T2 = T1	m·R·T·ln(V2/V1)
Polytropic	P·Vⁿ = const	(P2V2 − P1V1)/(1 − n)
General	Q or W given	from the balance

Worked example: constant-volume heating of air

A rigid tank holds m = 2 kg of air at P1 = 100 kPa, T1 = 300 K. It is heated to T2 = 500 K. Find Q.

Step 1 — the constraint. Constant volume ⇒ Wb = 0 (no boundary moves), so W = 0.

Step 2 — the balance. Q = W + m·(u2 − u1) = m·(u2 − u1).

Step 3 — get u from the air table. At 300 K, u1 = 210.49 kJ/kg; at 500 K, u2 = 359.49 kJ/kg.

Q = 2·(359.49 − 210.49) = 298.0 kJ

All the added heat raises internal energy; none leaves as work.

The constant-volume gotcha

For an ideal gas, u depends only on temperature, so T1 and T2 alone close both states. For water/steam, u depends on two properties, so a constant-volume process needs the specific volume carried from state 1 to state 2 — which means state 2 needs a second property (a pressure, a temperature, and the propagated volume). The closed-system calculator propagates mass and the process relation for you; give each state enough properties to close and it handles the bookkeeping.

Check yourself

Set the process to “Constant volume”, enter the two states, and the calculator reports Wb = 0 and Q = m·Δu. If you get a “need mass and internal energy at both states” warning, one of the states isn’t closed yet — add a property.