The Gibbs-Duhem Equation and Colligative Properties

Vocabulary and key equations regarding the Gibbs-Duhem equation, colligative properties including boiling/melting point changes, and osmotic pressure applications.

Gibbs-Duhem Equation

An equation demonstrating that at constant pressure ($p$) and temperature ($T$), the chemical potentials of a system's components do not vary independently, expressed as $\sum n_i d ext{μ}_i = 0$.

Chemical potential of an ideal solution ($ext{μ}_{A(l)}$)

The chemical potential of a solvent in an ideal mixture, defined by the equation $\text{μ}<em>{A(l)} = ext{μ}</em>{A(l)}^* + RT ext{ln}(x_A)$, which decreases as the mole fraction of solute ($x_B$) increases.

Colligative property

A property of a solution that depends only on the number of solute particles (mole fraction) present, rather than the chemical identity of those particles.

Boiling point elevation ($\text{Δ}T_b$)

The increase in a solvent's boiling point due to the addition of a solute, where $\text{Δ}T_b ∝ x_B$ and is specifically calculated as \text{Δ}T_b = rac{RT^{*2}x_B}{ ext{Δ}H_{vap}}.

Melting point depression ($\text{Δ}T_m$)

Also known as freezing point depression, it is the lowering of a solvent's melting point upon adding a solute, calculated as \text{Δ}T_m = rac{RT^{*2}x_B}{ ext{Δ}H_{fus}}.

Thermal stability range

The range between the melting and boiling points; it is greater for a solution than a pure liquid because a solute lowers the liquid's chemical potential while leaving the solid and gas phases typically unaffected.

Osmotic pressure (̠)

Defined as the pressure that, when applied to a solution, prevents the influx of solvent through a semi-permeable membrane.

van’t Hoff equation

An equation for the osmotic pressure of dilute solutions, given as \u0320 = rac{n_B RT}{V}, which shares the same form as the ideal gas equation.

Osmometry

A sensitive method for determining the molar masses of macromolecules in dilute solutions using osmotic pressure measurements, where M_{r,B} = rac{m_BRT}{̠ V}.

Isotonic

A condition where two solutions have the same osmotic pressure; solutions injected into the bloodstream must be isotonic with blood to prevent red blood cells from shrinking or swelling.

Reverse Osmosis

A water purification process (e.g., for sea water) where applied pressure forces solvent flow against the concentration gradient; it requires work greater than $-RT ext{ln}(x_A)$.

Entropy of the solution state

The thermodynamic factor that causes solute addition to lower the chemical potential ($ext{μ}$) and increase the thermal stability range, because the entropy of the solution is higher than that of the pure liquid.