Thermodynamics - Chemical Potential and Phase Equations

Flashcards covering chemical potential, thermodynamic stability, phase equilibrium criteria, and the derivation and application of the Clapeyron and Clausius-Clapeyron equations.

Chemical Potential ($\mu$)

Signifies the potential for change and is defined mathematically as $\mu = \left( \frac{\partial G}{\partial n} \right)_{p, T}$.

Gibbs free energy ($G$) extension for open systems

The relationship for an open system containing a single component is $dG = V dp - S dT + \mu dn$.

Extensive quantity

A quantity that is proportional to the size of the system, such as Gibbs free energy ($G$).

Intensive quantity

A quantity independent of system size, such as chemical potential ($\mu$), which has units of $J\,mol^{-1}$.

One-component system chemical potential identity

In a one-component system, the chemical potential is simply the molar Gibbs free energy: $\mu = G_m$.

Thermodynamic stability criterion (One-component)

The phase with the lowest chemical potential ($\mu$) will be thermodynamically stable at constant $T$ and $p$.

Triple point

The specific point on a phase diagram where three phases (gas, solid, and liquid) coexist at equilibrium.

Criterion for equilibrium

The chemical potential is the same in all phases present, formally expressed as $\mu^{(l)} = \mu^{(s)}$ for liquid-solid equilibrium.

Variation of $\mu$ with pressure ($p$)

Defined by the relationship $\left( \frac{\partial \mu}{\partial p} \right)_T = V_m$, where $V_m$ is the molar volume.

Molar Volume hierarchy

Usually, V_{m(g)} >> V_{m(l)} > V_{m(s)}, with water being a notable exception to the liquid-solid relationship.

Variation of $\mu$ with temperature ($T$)

Defined by the relationship $\left( \frac{\partial \mu}{\partial T} \right)_p = -S_m$, where $S_m$ is the molar entropy.

Molar Entropy hierarchy

The relationship between phases is S_{m(g)} >> S_{m(l)} > S_{m(s)}.

Clapeyron Equation (differential form)

Describes the slope of phase boundaries in $p$-$T$ diagrams: $\frac{dp}{dT} = \frac{\Delta_{tr} S}{\Delta_{tr} V} = \frac{\Delta_{tr} H}{T \Delta_{tr} V}$.

Clapeyron Equation (integrated form)

Assuming $\Delta H$ and $\Delta V$ do not vary over small temperature changes: $p_2 - p_1 = \frac{\Delta_{tr} H}{\Delta_{tr} V} \ln\left( \frac{T_2}{T_1} \right)$.

Clausius-Clapeyron Equation Assumptions

1) The vapour is a perfect gas ($V_m = \frac{RT}{p}$); 2) For liquid-vapour or solid-vapour transitions, V_{m(g)} >> V_{m(l)} so $\Delta_{tr} V \approx V_{m(g)}$.

Clausius-Clapeyron Equation (differential form)

Used when one phase is a vapour to describe boundaries where $\frac{d \ln(p/p^{\circ})}{dT} = \frac{\Delta_{tr} H}{RT^2}$.

Clausius-Clapeyron Equation (integrated form)

The relationship used to estimate vapour pressures or enthalpies of transition: $\ln\left( \frac{p_2}{p_1} \right) = -\frac{\Delta_{tr} H}{R} \left( \frac{1}{T_2} - \frac{1}{T_1} \right)$.

Maxwell relationship from $dG$

Because $dG$ is an exact differential, the order of operations does not matter, leading to $\left( \frac{\partial V}{\partial T} \right)_p = -\left( \frac{\partial S}{\partial p} \right)_T$.