Thermodynamics

Two systems each in thermal equilibrium with a third are in thermal equilibrium with each other. This is what makes a thermometer valid — touch each object to the same thermometer in turn, never to each other.

Q positive = heat absorbed by the system; W positive = work done by the system on its surroundings. Get a sign backwards and every later result flips.

= area under the P-V curve, path-dependent. $\Delta U$ is a state variable (endpoints only); Q absorbs whatever path-dependence is left over.

1. Isothermal (ΔT = 0, reservoir contact): Boyle’s law $PV = nRT = \text{constant}$; since $\Delta U = 0$, first law gives $Q = W$.
2. Adiabatic (Q = 0, insulated or fast): $PV^{\gamma} = \text{constant}$ (also $TV^{\gamma-1} = \text{constant}$); $\Delta U = -W$ — compression does work ON the gas with zero heat input, so T rises with no heater anywhere (diesel-engine effect).

$\gamma = C_p/C_v$ is always $> 1$, so the adiabatic curve through any point is always steeper than the isothermal curve through that same point.

Cp always exceeds Cv: the same ΔT costs more heat at constant pressure because some of it leaks out as expansion work instead of staying behind as ΔU.

Kelvin-Planck statement (second law): no process is possible whose sole result is absorbing heat from a reservoir and converting it completely into work — $\eta = 1$ is forbidden; every real engine rejects some heat to a cold sink.

T always in Kelvin. Four reversible steps between hot reservoir T₁ and cold reservoir T₂:

1. Isothermal expansion at T₁ — absorbs Q₁ from the hot reservoir.
2. Adiabatic expansion — cools T₁ → T₂, no heat exchange.
3. Isothermal compression at T₂ — rejects Q₂ to the cold reservoir.
4. Adiabatic compression — returns T₂ → T₁, closing the cycle.

These give $Q_2/Q_1 = T_2/T_1$. No engine beats $\eta_{Carnot}$ between the same two temperatures; it reaches 1 only if $T_2 = 0\text{ K}$ — unreachable (third law).

• 1 cal = 4.186 J
• 1 kPa × 1 L = 1 J — convenient shortcut for P-V work
• $\gamma = C_p/C_v$, always greater than 1 for an ideal gas
• T must be in Kelvin in every engine/Carnot formula