Each standard thermodynamic process holds one quantity constant, which simplifies the first law $Q = \Delta U + W$.
The volume stays constant, so the gas does no work: $W = 0$. All heat supplied changes the internal energy : $Q = \Delta U$. The pressure stays constant while the volume changes. The gas does work $W = P\Delta V$, so the first law gives $Q = \Delta U + P\Delta V$. The temperature stays constant, so for an ideal gas $\Delta U = 0$. All heat supplied is converted into work done by the gas: $Q = W$. No heat enters or leaves the system: $Q = 0$. Work done by the gas comes entirely from its internal energy: $\Delta U = -W$, so the gas cools as it expands. Do not confuse isothermal and adiabatic processes. Isothermal means constant temperature ($\Delta U = 0$); adiabatic means no heat transfer ($Q = 0$). The table summarises how the first law applies to each process.
Process Constant Quantity Key Equation Work Done Isovolumetric Volume ($V$) $Q = \Delta U$ $W = 0$ Isobaric Pressure ($P$) $Q = \Delta U + P \Delta V$ $W = P \Delta V$ Isothermal Temperature ($T$) $Q = W$ $W = nRT \ln \frac{V_2}{V_1}$ Adiabatic No heat exchange ($Q = 0$) $PV^{\gamma} = \text{constant}$ $W = \frac{P_1V_1 - P_2V_2}{\gamma - 1}$
During a reversible adiabatic process , pressure and volume for a monatomic ideal gas are related by: $$PV^{\frac{5}{3}} = \text{constant}$$ This comes from combining the ideal gas law with the first law under adiabatic conditions, with $\gamma = \tfrac{5}{3}$ for a monatomic gas. A gas undergoes adiabatic expansion from state $P_1$, $V_1$ to state $P_2$, $V_2$. The adiabatic equation gives: $$P_1V_1^{\frac{5}{3}} = P_2V_2^{\frac{5}{3}}$$ The Carnot cycle consists of four reversible processes. Isothermal Expansion (Process 1-2)Gas expands at constant temperature $T_h$. Absorbs heat $Q_h$ from the hot reservoir; work is done by the gas. Adiabatic Expansion (Process 2-3)No heat exchange; temperature drops from $T_h$ to $T_c$ as the gas does work. Isothermal Compression (Process 3-4)Gas is compressed at constant temperature $T_c$. Releases heat $Q_c$ to the cold reservoir; work is done on the gas. Adiabatic Compression (Process 4-1)No heat exchange; temperature rises from $T_c$ to $T_h$ as work is done on the gas. Pressure volume diagram of the Carnot cycle with isothermal and adiabatic expansion and compression. A heat engine converts thermal energy into mechanical work by operating in a repeating cyclic process.
It is defined as:
$$\eta = \frac{\text{Useful work}}{\text{Input energy}} = \frac{W}{Q_{\text{in}}}$$
A heat engine receives 500 J from the hot reservoir and performs 200 J of work. Its efficiency is: $$\eta = \frac{200 \, \text{J}}{500 \, \text{J}} = 0.4 \, \text{ or } \, 40\%$$ It is given by:
$$\eta_{\text{Carnot}} = 1 - \frac{T_c}{T_h}$$
For an engine operating between 600 K and 300 K, the Carnot efficiency is:
$$\eta_{\text{Carnot}} = 1 - \frac{300}{600} = 0.5 \, \text{ or } \, 50\%$$
The Carnot efficiency is an ideal upper limit set by the reservoir temperatures alone. It assumes perfectly reversible processes with no friction, turbulence, or heat loss. No real engine can reach or exceed it. Real engines, such as the Otto and Diesel engines, approximate but do not replicate the Carnot cycle. They have lower efficiencies because of:mechanical energy losses, non-reversible processes, and non-ideal working conditions, such as imperfect insulation. Describe the four processes that make up the Carnot cycle. What is the main difference between an isothermal and an adiabatic process? How is the efficiency of a heat engine calculated? Write the Carnot efficiency and explain why no real engine can exceed it. 
