Thermodynamics: Fundamental Equations, Entropy, and Gibbs Free Energy

Vocabulary practice flashcards covering entropy calculations, cyclic thermodynamic paths, fundamental Gibbs equations, Maxwell relations, Gibbs-Helmholtz equations, and the pressure/temperature dependence of thermodynamic potentials.

Entropy Change ($\Delta S$)

The thermodynamic property defined by $\Delta S = \int \frac{\text{d}q_{\text{rev}}}{T}$, which must be evaluated along a reversible path between the initial and final states; because entropy is a state function, $\Delta S$ has the same value for an irreversible path between those same states.

Constant-Volume Entropy Change of Heating

The entropy change for heating one mole of substance at constant volume, defined as $\Delta S = \int_{T_1}^{T_2} \frac{C_V}{T}\,\text{d}T$, which simplifies to $\Delta S = C_V \ln\left(\frac{T_2}{T_1}\right)$ when the constant-volume heat capacity $C_V$ is independent of temperature.

Constant-Pressure Entropy Change of Heating

The entropy change for heating one mole of substance at constant pressure, defined as $\Delta S = \int_{T_1}^{T_2} \frac{C_P}{T}\,\text{d}T$, where $C_P$ is the constant-pressure heat capacity of the substance.

Phase Change Entropy Change

The entropy change during a phase transition occurring at constant temperature and pressure, expressed as $\Delta S = \frac{\Delta H}{T}$, where $\Delta H$ represents the enthalpy of transition (such as the heat of vaporization).

Isothermal Entropy Change of an Ideal Gas

The entropy change for an ideal gas transitioning at constant temperature from $(P, V, T)$ to $(P', V', T)$, given by $\Delta S = R \ln\left(\frac{V'}{V}\right) = R \ln\left(\frac{P}{P'}\right)$, which is positive whenever the volume increases.

Path-Dependent Quantities

Thermodynamic quantities such as work ($w$) and heat ($q$) whose values depend on the pathway taken between states, characterized by non-zero cyclic integrals: $\oint D w \neq 0$ and $\oint D q \neq 0$.

State Function

A thermodynamic property whose change depends only on the initial and final states and whose cyclic integral over any closed path equals zero: $\oint \text{d}y = 0$.

Indicator Diagram

A pressure-volume diagram introduced by James Watt where the area under the trajectory represents the work done by the gas during an expansion or compression.

Maximum Work Principle

The thermodynamic principle stating that the work performed by a system in an expansion from state A to state B is maximized when carried out reversibly ($w_{\text{rev}} > w_{\text{irrev}}$), because pushing power is wasted when internal pressure exceeds external pressure.

Isothermal Reversible Expansion Work

The work done by an ideal gas during a reversible expansion at constant temperature, calculated as $w = -nRT \int_{V_i}^{V_f} \frac{\text{d}V}{V} = -nRT \ln\left(\frac{V_f}{V_i}\right)$.

Mechanical Properties

The macroscopic mechanical variables that define the physical dimensions and force state of a thermodynamic system: pressure ($P$) and volume ($V$).

Fundamental Thermodynamic Properties

The core thermodynamic variables directly defined by the fundamental laws of thermodynamics: temperature ($T$), internal energy ($U$), and entropy ($S$).

Composite Thermodynamic Properties

Properties formulated by combining fundamental and mechanical properties: enthalpy ($H = U + PV$), Helmholtz free energy ($A = U - TS$), and Gibbs free energy ($G = H - TS$).

Gibbs Equations (Four Fundamental Equations)

The four exact differential expressions for a closed system undergoing a reversible process with $P\text{--}V$ work only: $\text{d}U = T\text{d}S - P\text{d}V$, $\text{d}H = T\text{d}S + V\text{d}P$, $\text{d}A = -P\text{d}V - S\text{d}T$, and $\text{d}G = V\text{d}P - S\text{d}T$.

Maxwell Relations

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

Constant-Volume Heat Capacity ($C_V$)

The heat capacity defined at constant volume by the partial derivatives $C_V = \left(\frac{\partial U}{\partial T}\right)_V = T\left(\frac{\partial S}{\partial T}\right)_V$.

Constant-Volume Heat Capacity ($C_V$)

The heat capacity defined at constant volume by the partial derivatives $C_V = \left(\frac{\partial U}{\partial T}\right)_V = T\left(\frac{\partial S}{\partial T}\right)_V$.

Constant-Pressure Heat Capacity ($C_P$)

The heat capacity defined at constant pressure by the partial derivatives $C_P = \left(\frac{\partial H}{\partial T}\right)_P = T\left(\frac{\partial S}{\partial T}\right)_P$.

Gibbs-Helmholtz Equation

The thermodynamic relation $\left(\frac{\partial (G/T)}{\partial T}\right)_P = -\frac{H}{T^2}$ (or $\left(\frac{\partial (\Delta G/T)}{\partial (1/T)}\right)_P = \Delta H$), used to calculate enthalpy changes without calorimetry and to find $\Delta G$ at varying temperatures.

Gibbs-Helmholtz Plot

A graphical representation plotting $\frac{\Delta G}{T}$ against $\frac{1}{T}$, where the slope of the tangent to the curve equals the enthalpy change $\Delta H$.

Temperature Dependence of Gibbs Energy

The thermodynamic relationship $\left(\frac{\partial G}{\partial T}\right)_P = -S$, which indicates that an increase in temperature decreases the Gibbs free energy when pressure is kept constant.

Pressure Dependence of Gibbs Energy

The thermodynamic relationship $\left(\frac{\partial G}{\partial P}\right)_T = V$, which indicates that an increase in pressure increases the Gibbs free energy when temperature is kept constant.

Temperature Dependence of $G$ and $H$ Plot

A graphical plot of enthalpy $H$ and Gibbs energy $G$ versus temperature $T$ at constant pressure, illustrating that $G(0) = H(0)$, the slope of $H$ is $C_P$, the slope of $G$ is $-S$, and the vertical difference equals $TS$.

Gibbs Energy Pressure Dependence for Incompressible Phases

The relation $G_2 = G_1 + V(P_2 - P_1)$, valid for liquids and solids where the volume $V$ is nearly independent of pressure.

Gibbs Energy Pressure Dependence for an Ideal Gas

The equation $G = G^\circ + nRT \ln\left(\frac{P}{P^\circ}\right)$ or $\Delta G = nRT \ln\left(\frac{P_2}{P_1}\right)$, relating Gibbs free energy to pressure relative to a standard state pressure $P^\circ$.

Isothermal Free Expansion of an Ideal Gas

An irreversible expansion into a vacuum where $w = 0$, $\Delta U = 0$, and $q = 0$, but all state function changes ($\Delta H = 0$, $\Delta S$, $\Delta A$, and $\Delta G$) are identical to those of a reversible isothermal expansion between the same initial and final states.

Gibbs Free Energy Spontaneity Criteria

Spontaneity guidelines governed by $\Delta G = \Delta H - T\Delta S$: a transformation is always spontaneous if $\Delta H < 0$ and $\Delta S > 0$, never spontaneous if $\Delta H > 0$ and $\Delta S < 0$, and governed by the relative magnitudes of $\Delta H$ and $T\Delta S$ in other cases.