First Law of Thermodynamics

The Energy Ledger

The First Law of Thermodynamics is the universe's unbreakable accounting rule: a system's internal energy changes only by heat added or work done.

$$\Delta U = Q - W$$

Here, $\Delta U$ is internal energy change, $Q$ is heat added, and $W$ is work done by the system. A pump warms as you compress it (work becomes energy); a spray can chills as gas expands.

Internal energy for an ideal gas is $U = nC_V T$, and Mayer's relation links heat capacities by $C_P - C_V = R$.

Key insight: This law conserves energy but does not restrict direction—the second law handles that.

Ideal Gas Processes

Gas processes transform energy in distinct ways. The table below outlines the core behaviors:

For adiabatic processes, $PV^\gamma = \text{const}$ and $TV^{\gamma-1} = \text{const}$, where $\gamma = C_P/C_V$.

Common pitfall: $Q = 0$ does not mean $\Delta T = 0$. Adiabatic compression raises temperature purely through work—this is how a diesel engine fires without a spark plug.