First Law for a Closed Gas System (General Gas Energy Equation) Between states 1 and 2, the heat supplied equals the change in internal energy plus the work done: Q1–2 = dU + W1–2 (with consistent heat units).

Introduction / Context:
The general gas energy equation is simply the first law of thermodynamics written for a closed system. It states that heat added to a system is used to change its internal energy and to perform boundary work.

Given Data / Assumptions:

• Closed system of gas, undergoing any quasi-equilibrium process between states 1 and 2.
• Sign convention: heat added to the system and work done by the system are positive.
• Neglect kinetic and potential energy changes unless stated.

Concept / Approach:

The first law for a closed system is written as Q1–2 = ΔU + W1–2. Here, ΔU is U2 − U1 and W1–2 is boundary work (area under the p–v curve). The relation is independent of path specifics; it simply balances energy.

Step-by-Step Solution:

Verification / Alternative check:

Special cases: For an adiabatic process Q1–2 = 0 → W1–2 = −ΔU. For an isochoric process W1–2 = 0 → Q1–2 = ΔU. Both obey the same master equation.

Why Other Options Are Wrong:

Minus signs or algebraic operations like division or multiplication do not reflect energy conservation. The equality must be additive with proper signs.

Common Pitfalls:

Sign convention confusion (work by vs. on the system); forgetting to include boundary work only (not shaft work in closed simple systems unless explicitly present).

Final Answer:

Q1 - 2 = dU + W1 - 2