The Laws of Thermodynamics

1.1 Zeroth Law and Temperature

Zeroth Law: If system $A$ is in thermal equilibrium with system $B$And $B$ is in thermal equilibrium with system $C$Then $A$ is in thermal equilibrium with $C$.

This establishes temperature as a transitive equivalence relation: two systems are in thermal equilibrium if and only if they have the same temperature.

Definition. Temperature is the quantity that is equal for all systems in mutual thermal equilibrium. The ideal gas scale defines temperature via

$PV = Nk_BT$

Where $k_B = 1.381 \times 10^{-23}$ J/K is Boltzmann”s constant.

1.2 First Law

First Law: The change in internal energy of a system equals the heat added minus the work done by the system:

$dU = \delta Q - \delta W$

For a reversible process: $\delta W = P\,dV$ (PV work), giving

$dU = \delta Q - P\,dV$

Proposition 1.1. For an adiabatic process ($\delta Q = 0$): $dU = -P\,dV$. For an isochoric process ($dV = 0$): $dU = \delta Q$.

Definition. The heat capacity at constant volume and heat capacity at constant pressure are:

$C_V = \left(\frac{\partial U}{\partial T}\right)_V, \qquad C_P = \left(\frac{\partial H}{\partial T}\right)_P$

Where $H = U + PV$ is the enthalpy.

Proposition 1.2. For an ideal gas: $C_P - C_V = Nk_B$.

Proof. $H = U + PV = U + Nk_BT$. Therefore $C_P = (\partial H/\partial T)_P = (\partial U/\partial T)_P + Nk_B = C_V + Nk_B$ (since $U$ depends only on $T$ for an ideal gas). $\blacksquare$

1.3 Second Law and Entropy

Second Law (Clausius statement): Heat cannot spontaneously flow from a colder body to a hotter body.

Second Law (Kelvin-Planck statement): No process can convert heat entirely into work in a cyclic manner without other effects.

These are equivalent: each implies the other.

Definition. The entropy change for a reversible process is

dS = \frac{\delta Q_{\mathrm{rev}}{T}}

Theorem 1.3 (Clausius Inequality). For any cyclic process:

$\oint \frac{\delta Q}{T} \leq 0$

With equality for reversible processes.

Proof. Consider a system undergoing a cycle interacting with $n$ heat reservoirs at temperatures $T_1, \ldots, T_n$Exchanging heat $Q_i$ with reservoir $i$. The Clausius inequality follows from the impossibility of a perpetual motion machine of the second kind: a cycle that absorbs heat from a single reservoir and does work would violate the Kelvin-Planck statement. The detailed …/1-number-and-algebra/3_proof-and-logic uses auxiliary Carnot engines operating between pairs of reservoirs. $\blacksquare$

Corollary 1.4 (Principle of Increasing Entropy). For an isolated system, $dS \geq 0$With equality for reversible processes.

1.4 Third Law

Third Law (Nernst): As $T \to 0^+$The entropy of a perfect crystal approaches a constant (which can be taken as zero):

$\lim_{T \to 0} S(T) = 0$

Consequences:

1. It is impossible to reach absolute zero in a finite number of steps.
2. The heat capacities $C_V$ and $C_P$ approach zero as $T \to 0$.

1.5 Thermodynamic Potentials

Theorem 1.5. At equilibrium for a system in contact with a heat bath at temperature $T$: $F$ is minimised at constant $T, V$; $G$ is minimised at constant $T, P$.

Proof. For constant $T, V$: $dF = dU - TdS = \delta Q - PdV - TdS$. At equilibrium $dS \geq \delta Q/T$ (Clausius inequality), so $dF \leq 0$. Hence $F$ decreases and is minimised at equilibrium. The argument for $G$ is analogous. $\blacksquare$

1.6 Maxwell Relations

From the exactness of $dU = TdS - PdV$ (and similarly for $dH$, $dF$, $dG$), the equality of mixed partial derivatives gives four Maxwell relations:

1. $\left(\frac{\partial T}{\partial V}\right)_S = -\left(\frac{\partial P}{\partial S}\right)_V$
2. $\left(\frac{\partial T}{\partial P}\right)_S = \left(\frac{\partial V}{\partial S}\right)_P$
3. $\left(\frac{\partial S}{\partial V}\right)_T = \left(\frac{\partial P}{\partial T}\right)_V$
4. $\left(\frac{\partial S}{\partial P}\right)_T = -\left(\frac{\partial V}{\partial T}\right)_P$

1.7 Common Pitfalls

• $\delta Q$ and $\delta W$ are not exact differentials. Unlike $dU$The heat and work are path-dependent. Only $\delta Q_{\mathrm{rev}/T = dS}$ is exact.
• The second law prohibits certain processes but does not explain why they occur. Statistical mechanics provides the microscopic explanation: entropy measures the number of microstates, and the system evolves toward the macrostate with the most microstates.
• Free energy minima determine equilibrium, not energy minima. At constant temperature, the system minimises $F$ (or $G$), not $U$.