First Law of Thermodynamics

Energy balance between heat, work, and internal energy change in a system.

Conceptual context

The first law of thermodynamics is an energy-conservation statement for a thermodynamic system. It does not say that heat and work are stored inside matter. It says that energy can cross the system boundary as $Q$ or $W$, and that the net effect inside the system is $\Delta U$. The boundary is therefore the central object of the model: it decides what belongs to the system and what belongs to the surroundings. Once that boundary is fixed, the question becomes operational: how do heat, work, and internal energy change combine to describe the same physical process?

🟢 Essential level

The first law is an energy account book for a closed system. If energy enters as $Q$ or as $W$, the system gains internal energy. If energy leaves through either route, the system loses internal energy. The key distinction is between transfer and storage: the system stores internal energy, not heat or work. Heat is transfer caused by a temperature difference; work is transfer caused by mechanical action such as compression, expansion, pushing, or deformation.

$\Delta U$ is the result of the accounting. A positive value means the final state contains more internal energy than the initial state; a negative value means the opposite. Therefore every problem must begin by fixing the sign convention before adding all process terms consistently and carefully.

Think of a bank account: the final balance does not distinguish whether the money came in as one large deposit and one small withdrawal, or as several smaller deposits and withdrawals that cancel out. Only the net sum matters for the balance. The same holds for internal energy: $Q$ and $W$ can combine in many different ways over the course of a process, but only their net sum determines how much $\Delta U$ changed.

🔵 Formal level

Basic formulas

This leaf uses the convention of work done on the system. With that convention, $Q$ is positive when heat enters the system, and $W$ is positive when the surroundings do work on the system. The central relation is:

First law with work done on the system

This formula does not mean that heat or work are state variables. It means that both are transfer mechanisms that modify internal energy. Internal energy itself is a state function, so its change is defined by comparing the final state with the initial state:

Internal energy change between states

Real student questions

If heat enters, does temperature always rise? Not always. $Q$ may enter while the system does work on the surroundings, so part of the transferred energy does not remain as internal energy.

Why does work have the opposite sign in some books? Because some texts define positive work as work done by the system. This leaf uses $W$, positive when the surroundings work on the system.

Can $\Delta U$ be zero even if heat is exchanged? Yes. In a cycle, or in a process where incoming heat is compensated by outgoing work, the final internal state may equal the initial state.

Final synthesis

The first law is the energy accounting of a closed system. $Q$ and $W$ cross the boundary; $\Delta U$ remains as the internal result. Understanding signs, units, and system boundary is understanding the law.