Thermodynamics — First Law & Processes (Medium)

Quick Recap — First Law & Processes

• First law: $Q=\Delta U+W$, where $W=\int P\,dV$ is the work done by the gas. Internal energy of an ideal gas depends only on temperature: $\Delta U=nC_V\Delta T$.
• Process work: isochoric $W=0$; isobaric $W=P\,\Delta V=nR\,\Delta T$; isothermal $W=nRT\ln\dfrac{V_2}{V_1}$; adiabatic $Q=0$ so $\Delta U=-W$.
• Molar heats: $C_P-C_V=R$, $\gamma=\dfrac{C_P}{C_V}$, $C_V=\dfrac{R}{\gamma-1}$. Monatomic $\gamma=\tfrac53$, diatomic $\gamma=\tfrac75$.
• Adiabatic: $PV^\gamma=$ const and $TV^{\gamma-1}=$ const; on a $P$–$V$ diagram an adiabat is steeper than an isotherm.
• Cyclic process: $\Delta U=0$, so $Q=W=$ area enclosed (positive if clockwise).

Worked mini-example. 100 J of heat is supplied to a gas that does 40 J of work. By the first law $\Delta U=Q-W=100-40=60$ J — its temperature rises.