Review

1. Checklist of Key Concepts¶

Section 3.1: Conservation of Energy¶

1. First Law of Thermodynamics

   • The internal energy change of a system equals the heat absorbed plus the work done on it:

   $$\Delta U = q + w.$$

   (1)

   • In differential form: $dU = \delta q + \delta w$, or equivalently $\delta q = dU - \delta w$.

2. State Functions vs. Path Functions

   • $U$ is a state function: $\Delta U$ depends only on the initial and final states, and $dU$ is an exact differential ($\oint dU = 0$).

   • $q$ and $w$ are path functions: their values depend on the process path, and $\delta q$, $\delta w$ are inexact differentials.

3. Sign Convention (Chemistry Convention)

   • $q > 0$: heat absorbed by the system.

   • $w > 0$: work done on the system.

   • For $PV$ work: $\delta w = -P\,dV$, so compression ($dV < 0$) gives $w > 0$.

4. Generalized Work

   • Many work modes share a generalized force $\times$ generalized displacement structure: $PV$ work ($-P\,dV$), surface work ($\gamma\,dA$), elastic work ($k\,dl$), electrical work ($\mathcal{E}\,dq_{\mathrm{el}}$), and chemical work ($\mu\,dN$).

5. Free Expansion

   • Expansion into vacuum: $P_{\mathrm{ext}} = 0$, so $w = 0$. If the container is also insulated, $q = 0$ and $\Delta U = 0$. For an ideal gas, $\Delta T = 0$ as well.

   • Free expansion is the extreme case of an irreversible process: the gas changes state while doing no work and exchanging no heat.

Section 3.2: Applications of the First Law¶

1. Thermodynamic Processes

   • Quasi-static: carried out infinitesimally slowly; the system remains near equilibrium at all times.

   • Reversible: quasi-static and free of dissipative effects (friction, turbulence, viscous drag, etc.); can be reversed with no net change to system or surroundings.

   • Irreversible: any real process that violates one or more conditions for reversibility.

2. Reversible Work Bounds

   • For a given expansion ($\Delta V > 0$), a reversible process does the maximum work on the surroundings ($|w_{\mathrm{rev}}| \geq |w_{\mathrm{irrev}}|$).

   • For a given compression, a reversible process requires the minimum work input.

3. Five-Step First Law Workflow

   1. Choose two independent variables (e.g., $V, T$ or $P, T$).

   2. Rewrite the First Law in terms of those variables.

   3. Apply process constraints (isothermal, isochoric, isobaric, adiabatic).

   4. Specify the equation of state (e.g., ideal gas).

   5. Integrate to find $q$, $w$, $\Delta U$.

4. First Law with $V$ and $T$ as Independent Variables

   • General form:

   $$\delta q \;=\; C_V \, dT \;+\; \left[\left(\frac{\partial U}{\partial V}\right)_T + P\right] dV.$$

   (2)

   • For an ideal gas, $(\partial U/\partial V)_T = 0$, simplifying to $\delta q = C_V\,dT + P\,dV$.

5. Common Ideal-Gas Processes

   • Isochoric ($dV = 0$): $w = 0$, $q = C_V\,\Delta T$, $\Delta U = C_V\,\Delta T$.

   • Isothermal ($dT = 0$): $\Delta U = 0$ (ideal gas), $q = -w = Nk_{\mathrm{B}}T\ln(V_2/V_1)$.

   • Adiabatic ($\delta q = 0$): $\Delta U = w = C_V\,\Delta T$, and $TV^{\gamma - 1} = \text{const}$ (equivalently $PV^\gamma = \text{const}$), where $\gamma = C_P/C_V$.

6. Microscopic Interpretation of the First Law

   • From $U = \sum_i p_i E_i$:

   $$dU = \underbrace{\sum_i E_i\, dp_i}_{\delta q} + \underbrace{\sum_i p_i\, dE_i}_{\delta w}.$$

   (3)

   • Heat changes which states are occupied (probabilities shift; energy levels fixed).

   • Work changes the energies of the states (energy levels shift; probabilities fixed).

Section 3.3: Enthalpy¶

1. Definition and Motivation

   • Enthalpy: $H = U + PV$.

   • At constant pressure with $PV$-only work: $\delta q_P = dH$, so $q_P = \Delta H$.

   • Defines a state function that plays the same role at constant $P$ that $U$ plays at constant $V$.

2. Heat Capacity at Constant Pressure

   $$C_P = \left(\frac{\partial H}{\partial T}\right)_P = \left(\frac{\partial U}{\partial T}\right)_P + P\left(\frac{\partial V}{\partial T}\right)_P.$$

   (4)

   • For an ideal gas: $C_P - C_V = Nk_{\mathrm{B}}$ (or $nR$ per mole), so $C_P > C_V$.

3. When $q_P = \Delta H$ Fails

   • The relation holds only when $PV$ work is the sole form of work. Non-$PV$ work at constant pressure (e.g., electrical work in an electrochemical cell) breaks the equivalence.

4. Standard States and Formation Enthalpies

   • Standard pressure: $P^\circ = 1\,\text{bar}$; reference temperature typically 298.15 K.

   • Standard enthalpy of formation, $\Delta H_f^\circ$: enthalpy change when 1 mol of compound is formed from elements in their standard states.

   • By convention, $\Delta H_f^\circ = 0$ for elements in their standard states.

5. Hess’s Law

   • Because $H$ is a state function, enthalpy changes are path-independent and additive:

   $$\Delta H_{\mathrm{rxn}}^\circ = \sum_{p} \nu_p \,\Delta H_{f,p}^\circ - \sum_{r} \nu_r \,\Delta H_{f,r}^\circ.$$

   (5)

6. First Law Toolkit Summary

   Constraint	Relevant state function	Key relation
   Constant $V$	$U$	$q_V = \Delta U$
   Constant $P$	$H = U + PV$	$q_P = \Delta H$

2. Checklist of Most Important Equations¶

Below is a unified list of the major equations from Sections 3.1–3.3.

A. First Law of Thermodynamics

• Applicability: any closed system. $q > 0$ = heat absorbed by the system; $w > 0$ = work done on the system.

B. $PV$ Work

• Applicability: any expansion or compression. For reversible processes, $P_{\mathrm{ext}} = P$ (system pressure). For irreversible processes, $P_{\mathrm{ext}}$ is determined by the surroundings.

C. First Law in $(V, T)$ Variables (for $PV$-only work)

• For an ideal gas, $(\partial U/\partial V)_T = 0$, giving $\delta q = C_V\,dT + P\,dV$.

D. Isothermal Reversible Work (ideal gas)

• $\Delta U = 0$ for an ideal-gas isothermal process.

E. Adiabatic Relations (ideal gas, reversible)

• For a monatomic ideal gas, $\gamma = 5/3$ and the $TV$ relation becomes $TV^{2/3} = \text{const}$.

F. Isochoric Process ($dV = 0$)

G. Enthalpy

• At constant pressure with $PV$-only work: $q_P = \Delta H$.

H. Heat Capacity at Constant Pressure

• For an ideal gas: $C_P - C_V = Nk_{\mathrm{B}}$ (per particle) or $C_P - C_V = nR$ (per mole).

I. Hess’s Law

• Applicability: any reaction at standard conditions, using tabulated formation enthalpies.

J. Microscopic First Law

• Applicability: closed system with $PV$-only work. Heat redistributes probabilities; work shifts energy levels.