Thermodynamics | Chemistry - Wyatt's Notes

1. Zeroth Law: If $A$ is in thermal equilibrium with $B$, and $B$ with $C$, then $A$ is in thermal equilibrium with $C$. This establishes temperature as a transitive property and justifies the use of thermometers.

2. First Law: Energy is conserved. For a closed system:

   $dU = \delta q + \delta w$

   where $U$ is internal energy, $q$ is heat, and $w$ is work. The notation $\delta$ indicates inexact differentials: $q$ and $w$ are path-dependent, but $U$ is a state function.

For a reversible expansion of an ideal gas against an external pressure:

$\delta w_{\text{rev}} = -P\,dV$

$w_{\text{rev}} = -\int_{V_i}^{V_f} P\,dV = -nRT\ln\frac{V_f}{V_i}$

Definition 1 (Enthalpy): The enthalpy $H$ is defined as:

$H = U + PV$

For a process at constant pressure:

$dH = dU + P\,dV + V\,dP = \delta q_p + V\,dP \implies \Delta H = q_p$

The molar heat capacities relate to enthalpy and internal energy:

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

For an ideal gas: $C_p - C_V = nR$.

Theorem 1 (Clausius Inequality): For any cyclic process:

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

Equality holds only for reversible processes. This implies the existence of a state function $S$ (entropy) such that:

$dS \geq \frac{\delta q}{T}$

For a spontaneous (irreversible) process in an isolated system: $dS > 0$.

For a reversible process at temperature $T$:

$\Delta S = \int_{T_1}^{T_2} \frac{C}{T}\,dT$

Entropy of phase transition: At the transition temperature $T_{\text{trs}}$:

$\Delta_{\text{trs}}S = \frac{\Delta_{\text{trs}}H}{T_{\text{trs}}}$

Example 1: Calculate $\Delta S$ when 2 mol of ice melts at 273 K ($\Delta_{\text{fus}}H = 6.01$ kJ/mol).

$\Delta S = \frac{n\,\Delta_{\text{fus}}H}{T} = \frac{2 \times 6010}{273} = 44.0 \text{ J/K}$

$\blacksquare$

Theorem 2 (Boltzmann Entropy):

$S = k_B \ln W$

where $W$ is the number of microstates and $k_B = 1.381 \times 10^{-23}$ J/K is Boltzmann”s constant.

For $N$ distinguishable particles with $n_i$ in each energy level $\varepsilon_i$:

$W = \frac{N!}{n_1!\,n_2!\,\cdots}$

The entropy of mixing two ideal gases:

$\Delta_{\text{mix}}S = -nR\left(x_A \ln x_A + x_B \ln x_B\right)$

Theorem 3 (Third Law of Thermodynamics): The entropy of a perfect crystal at absolute zero is zero:

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

This provides a reference point for absolute entropies (standard molar entropies $S^\circ$).

Definition 2 (Helmholtz Free Energy):

$A = U - TS$

$dA = -S\,dT - P\,dV$

Definition 3 (Gibbs Free Energy):

$G = H - TS = U + PV - TS$

$dG = -S\,dT + V\,dP$

At constant $T$ and $P$: $\Delta G = w_{\text{non-PV}}$, so the Gibbs free energy change equals the maximum non-expansion work.

The four fundamental equations of thermodynamics (for closed systems of constant composition):

$dU = T\,dS - P\,dV$ $dH = T\,dS + V\,dP$ $dA = -S\,dT - P\,dV$ $dG = -S\,dT + V\,dP$

Definition 4 (Chemical Potential): For an open system with $k$ components:

$dG = -S\,dT + V\,dP + \sum_{i=1}^{k} \mu_i\,dn_i$

where $\mu_i = \left(\frac{\partial G}{\partial n_i}\right)_{T,P,n_{j\neq i}}$ is the chemical potential of component $i$.

For an ideal gas: $\mu = \mu^\circ + RT\ln\frac{P}{P^\circ}$.

Theorem 4 (Maxwell Relations): Since $U$, $H$, $A$, $G$ are state functions, their mixed second partial derivatives are equal:

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

Using the Maxwell relation $\left(\frac{\partial S}{\partial V}\right)_T = \left(\frac{\partial P}{\partial T}\right)_V$:

For an ideal gas: $\left(\frac{\partial P}{\partial T}\right)_V = \frac{nR}{V}$, so:

$\Delta S = \int_{V_1}^{V_2} \frac{nR}{V}\,dV = nR\ln\frac{V_2}{V_1}$

Theorem 5 (Gibbs-Helmholtz Equation):

$\left[\frac{\partial(G/T)}{\partial T}\right]_P = -\frac{H}{T^2}$

Equivalently:

$\frac{\Delta G_2}{T_2} - \frac{\Delta G_1}{T_1} = -\Delta H\left(\frac{1}{T_2} - \frac{1}{T_1}\right)$

(approximate form when $\Delta H$ is constant over the temperature range).

At phase equilibrium between two phases $\alpha$ and $\beta$:

$\mu_\alpha(T, P) = \mu_\beta(T, P)$

Differentiating along the coexistence curve:

$d\mu_\alpha = d\mu_\beta \implies -S_\alpha\,dT + V_\alpha\,dP = -S_\beta\,dT + V_\beta\,dP$

$\frac{dP}{dT} = \frac{\Delta_{\text{trs}}S}{\Delta_{\text{trs}}V} = \frac{\Delta_{\text{trs}}H}{T\,\Delta_{\text{trs}}V}$

For liquid-vapor equilibrium, assuming $\Delta_{\text{vap}}H$ is constant and $V_g \gg V_l$:

$\ln\frac{P_2}{P_1} = -\frac{\Delta_{\text{vap}}H}{R}\left(\frac{1}{T_2} - \frac{1}{T_1}\right)$

Example 2: The normal boiling point of benzene is 353 K with $\Delta_{\text{vap}}H = 30.8$ kJ/mol. Find the vapor pressure at 298 K.

$\ln\frac{P}{1.013 \times 10^5} = -\frac{30800}{8.314}\left(\frac{1}{298} - \frac{1}{353}\right) = -1.93$

$P = 1.013 \times 10^5 \times e^{-1.93} = 1.47 \times 10^4 \text{ Pa} \approx 14.7 \text{ kPa}$

$\blacksquare$

Theorem 6 (Gibbs Phase Rule): For a system with $C$ components and $P$ phases at equilibrium:

$F = C - P + 2$

where $F$ is the number of degrees of freedom (intensive variables that can be independently varied).

For a single-component system ($C = 1$): $F = 3 - P$. At a triple point ($P = 3$), $F = 0$.

• Triple point: All three phases coexist; $F = 0$.
• Critical point: Termination of the liquid-vapor coexistence curve; above this point the fluid is supercritical.
• Slope of solid-liquid boundary: Positive for most substances (liquid is denser); negative for water (ice is less dense).

For binary mixtures, common diagrams include:

• Temperature-composition diagrams for liquid-vapor equilibrium (distillation).
• Eutectic diagrams for solid-liquid equilibrium.
• Lever rule: Determines the mass fractions of phases in a two-phase region.

Definition 5 (Lever Rule): For a two-phase region with phases $\alpha$ and $\beta$ at overall composition $x$:

$\frac{n_\alpha}{n_\beta} = \frac{x_\beta - x}{x - x_\alpha}$

At equilibrium, $\Delta_r G = 0$, giving:

$\Delta_r G^\circ = -RT\ln K$

For the reaction $aA + bB \rightleftharpoons cC + dD$:

$K = \frac{a_C^c\,a_D^d}{a_A^a\,a_B^b}$

where $a_i$ are activities. For ideal gases: $a_i = P_i/P^\circ$, so:

$K_p = \frac{(P_C/P^\circ)^c\,(P_D/P^\circ)^d}{(P_A/P^\circ)^a\,(P_B/P^\circ)^b}$

Theorem 7 (van’t Hoff Equation): The temperature dependence of the equilibrium constant:

$\frac{d\ln K}{dT} = \frac{\Delta_r H^\circ}{RT^2}$

Integrated form (assuming $\Delta_r H^\circ$ is constant):

$\ln\frac{K_2}{K_1} = -\frac{\Delta_r H^\circ}{R}\left(\frac{1}{T_2} - \frac{1}{T_1}\right)$

Definition 6 (Le Chatelier’s Principle): If a system at equilibrium is subjected to a disturbance, the system shifts to partially counteract the change.

• Increasing $T$ favors the endothermic direction.
• Increasing $P$ favors the direction with fewer moles of gas.
• Adding a reactant shifts equilibrium toward products.

Theorem 8 (Hess’s Law): The enthalpy change for a reaction is independent of the pathway; it equals the sum of enthalpy changes for any series of steps into which the reaction can be divided.

$\Delta_r H = \sum \Delta_f H^\circ(\text{products}) - \sum \Delta_f H^\circ(\text{reactants})$

• Standard enthalpy of formation: $\Delta_f H^\circ$ — enthalpy change when 1 mol of compound forms from its elements in their standard states.
• Standard enthalpy of combustion: $\Delta_c H^\circ$ — enthalpy change for complete combustion of 1 mol of substance.
• Bond enthalpies: Average energy required to break a bond in the gas phase.

Theorem 9 (Kirchhoff’s Law): Temperature dependence of reaction enthalpy:

$\frac{d\Delta_r H^\circ}{dT} = \Delta_r C_p^\circ$

Definition 7 (Partial Molar Volume): The partial molar volume of component $i$:

$V_i = \left(\frac{\partial V}{\partial n_i}\right)_{T,P,n_{j\neq i}}$

The total volume of a mixture:

$V = \sum_i n_i V_i$

Theorem 10 (Gibbs-Duhem Equation): At constant $T$ and $P$:

$\sum_i n_i\,d\mu_i = 0$

For a binary mixture: $n_A\,d\mu_A + n_B\,d\mu_B = 0$.

For a real solution, the chemical potential is:

$\mu_i = \mu_i^\circ + RT\ln a_i = \mu_i^\circ + RT\ln(\gamma_i\,x_i)$

where $\gamma_i$ is the activity coefficient and $x_i$ is the mole fraction. For ideal solutions ($\gamma_i = 1$):

$\mu_i = \mu_i^\circ + RT\ln x_i$

Theorem 11 (Carnot Efficiency): A heat engine operating between hot reservoir $T_h$ and cold reservoir $T_c$:

$\eta = 1 - \frac{T_c}{T_h}$

This is the maximum possible efficiency for any engine operating between these temperatures.

• Coefficient of Performance (refrigerator): $\text{COP}_{\text{ref}} = \frac{T_c}{T_h - T_c}$
• Coefficient of Performance (heat pump): $\text{COP}_{\text{hp}} = \frac{T_h}{T_h - T_c}$

For a real gas undergoing throttling (isenthalpic expansion):

$\mu_{JT} = \left(\frac{\partial T}{\partial P}\right)_H = \frac{1}{C_p}\left[\frac{2a}{RT} - b\right]$

For an ideal gas: $\mu_{JT} = 0$ (no temperature change on throttling).

For a reversible adiabatic process with an ideal gas ($\gamma = C_p/C_V$):

$TV^{\gamma-1} = \text{const}, \quad PV^\gamma = \text{const}$

Work done:

$w = \frac{nR(T_2 - T_1)}{\gamma - 1}$

Definition 8 (Fugacity): For a real gas:

$\mu = \mu^\circ + RT\ln\left(\frac{f}{P^\circ}\right)$

where $f = \phi P$ and $\phi$ is the fugacity coefficient. As $P \to 0$, $f \to P$ and $\phi \to 1$.

For condensed phases:

$a_i = \frac{f_i}{f_i^\circ} \approx \gamma_i\,x_i$

The equilibrium constant in terms of activities:

$K = \prod_i a_i^{\nu_i}$

1. Confusing heat ($q$) and temperature ($T$). Heat is energy in transit due to a temperature difference; temperature is a state property. Fix: $q$ is path-dependent; $T$ is a state function. Use $dU = \delta q + \delta w$, not $dU = T\,dS + \ldots$ for irreversible processes.
2. Using $\Delta G < 0$ as the sole spontaneity criterion. This only applies at constant $T$ and $P$. Fix: Use $dA < 0$ at constant $T$, $V$, or $dS > 0$ for isolated systems.
3. Ignoring the standard state. $\Delta G^\circ$ and $K$ are related, but $\Delta G = \Delta G^\circ + RT\ln Q$, where $Q$ is the reaction quotient. Fix: Only at equilibrium does $\Delta G = 0$ and $Q = K$.
4. Assuming $\Delta H$ and $\Delta S$ are temperature-independent. This is an approximation valid only over small temperature ranges. Fix: Use Kirchhoff’s law or integrate $C_p$ data when precision is needed.
5. Confusing intensive and extensive properties. $G$ is extensive; $\mu = G/n$ is intensive. Fix: Always use molar quantities when comparing substances with different amounts.
6. Wrong sign in the Clausius-Clapeyron equation. The negative sign appears because $\ln(P)$ decreases as $1/T$ increases for exothermic vaporization. Fix: Write it as $\ln(P_2/P_1) = -(\Delta H/R)(1/T_2 - 1/T_1)$ and check units.
7. Applying the ideal gas law to phase equilibrium without correction. The integrated Clausius-Clapeyron equation assumes $V_g \gg V_l$ and ideal gas behavior. Fix: Use fugacity corrections for high-pressure systems.

flowchart TD
    A[Thermodynamics] --> B[Key Concepts]
    A --> C[Core Principles]
    A --> D[Practical Applications]
    B --> E[Fundamental definitions]
    C --> F[Design patterns]
    D --> G[Real-world usage]

• First Law: $dU = \delta q + \delta w$; energy conservation.
• Second Law: $dS \geq \delta q/T$; entropy always increases in isolated systems.
• Third Law: $S \to 0$ as $T \to 0$ for a perfect crystal.
• Gibbs free energy: $\Delta G = \Delta H - T\Delta S$; spontaneity criterion at constant $T$, $P$.
• Chemical potential: $\mu_i = (\partial G/\partial n_i)_{T,P}$; drives mass transfer and chemical equilibrium.
• Maxwell relations: Connect measurable quantities derived from exact differentials of state functions.
• Phase rule: $F = C - P + 2$; determines degrees of freedom at equilibrium.
• Clausius-Clapeyron: $\ln(P_2/P_1) = -(\Delta H/R)(1/T_2 - 1/T_1)$; describes vapor pressure vs temperature.
• Equilibrium: $\Delta_r G^\circ = -RT\ln K$; van’t Hoff equation for temperature dependence.

Problem: The boiling point of water is 100 degrees C at 1 atm. The enthalpy of vaporization is 40.7 kJ/mol. Calculate the boiling point at 0.8 atm. Solution: ln(P2/P1) = -(Delta H_vap/R)(1/T2 - 1/T1). ln(0.8/1.0) = -(40700/8.314)(1/T2 - 1/373). -0.2231 = -4893(1/T2 - 0.00268). 1/T2 = 0.00268 + 0.2231/4893 = 0.00268 + 4.56e-5 = 0.002726. T2 = 366.8 K = 93.7 degrees C.

Problem: For the reaction N2(g) + 3H2(g) -> 2NH3(g), Delta H = -92.4 kJ/mol, Delta S = -198.8 J K^-1 mol^-1. At 298 K, determine if the reaction is spontaneous. Solution: Delta G = Delta H - T Delta S = -92,400 - 298(-198.8) = -92,400 + 59,200 = -33,200 J/mol = -33.2 kJ/mol. Delta G < 0, so the reaction is spontaneous at 298 K. At what T does it become non-spontaneous? Delta G = 0 when T = Delta H/Delta S = 92,400/198.8 = 464.8 K.

Problem: Calculate the total entropy change when 1.00 mol of water boils at 373 K and 1 atm. The enthalpy of vaporization is 40.7 kJ/mol. What is the entropy change of the surroundings?

Solution: For a reversible phase transition at constant T and P: Delta S_sys = Delta H_vap/T = 40,700/373 = 109.1 J K^-1 mol^-1. The surroundings lose the same heat: Delta S_surr = -Delta H_vap/T = -109.1 J K^-1 mol^-1. Delta S_univ = Delta S_sys + Delta S_surr = 0 for a reversible process at equilibrium (boiling at the normal boiling point is an equilibrium process).

Common mistake: Forgetting that Delta G = 0 at the boiling point, not Delta H = 0. The boiling point is where liquid and vapor are in equilibrium. Also, the entropy of the surroundings changes in the opposite direction to the system: when the system absorbs heat, the surroundings lose it.

$\blacksquare$

Thermodynamics tells us which processes can happen and how much energy they involve. Think of enthalpy as a heat budget: exothermic reactions release heat (like spending money), while endothermic reactions absorb it (like earning money). But enthalpy alone does not decide spontaneity. Entropy is the wild card, nature’s tendency toward disorder and the spreading of energy. A reaction is spontaneous when the free energy balance favors it, combining both the heat exchange and the entropy change. The second law says that in any process, the total entropy of the universe must increase. Phase transitions happen at exact temperatures because that is where the enthalpy and entropy contributions exactly balance. The Clausius-Clapeyron equation connects vapor pressure to temperature, explaining why liquids boil at higher temperatures under pressure: you are forcing the equilibrium point upward. Chemical equilibrium is not stasis but a dynamic balance where forward and reverse reactions proceed at equal rates, and Le Chatelier’s principle describes how the system nudges itself back when disturbed, like a thermostat adjusting to temperature changes.

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