First Law of Thermodynamics

The first law is the universe’s unbreakable accounting rule: a system’s internal energy changes only by heat in or work out. A bicycle pump warms as you compress (work becomes internal energy); a spray can chills as gas expands (internal energy becomes work). No process ever cheats the ledger.

$$\Delta U = Q - W$$

where $\Delta U$ is internal energy change, $Q$ is heat added, $W$ is work done by the system.

Work by a gas: $W = \int_{V_i}^{V_f} P\,dV$

Ideal gas processes

• Isothermal ($T$ const): $W = nRT\ln(V_f/V_i)$, $Q = W$
• Isobaric ($P$ const): $W = P\Delta V$, $Q = nC_P\Delta T$
• Isochoric ($V$ const): $W = 0$, $Q = nC_V\Delta T$
• Adiabatic ($Q = 0$): $W = -\Delta U$

Internal energy: $U = nC_V T$

Mayer's relation: $C_P - C_V = R$

Adiabatic relations:

$$PV^\gamma = \text{const}, \quad TV^{\gamma-1} = \text{const}$$

where $\gamma = C_P/C_V$.

Key insight: The first law conserves energy but doesn't say which processes are possible — that requires the second law.

Common pitfall: $Q = 0$ (no heat exchanged) does not mean $\Delta T = 0$. Adiabatic compression raises temperature purely through work — that is how a diesel engine ignites fuel with no spark plug at all.