Freezing Point Depression — Theory
When a non-volatile solute is dissolved in a solvent, the solution’s freezing point drops below that of the pure solvent.
For sufficiently dilute solutions, this colligative effect depends only on the number of dissolved particles, not their identity.
Core relations
The quantitative relation is the van ’t Hoff (cryoscopic) equation:
\[
\Delta T_f \;=\; i\,K_f\,m
\]
- \(\Delta T_f = T_f^{\circ} - T_f\) is the freezing point depression (°C or K; the magnitude is the same in either unit).
- \(i\) is the van ’t Hoff factor (effective number of dissolved particles per formula unit; \(i=1\) for nonelectrolytes; \(i\lt\) theoretical if there is ion pairing/association).
- \(K_f\) is the cryoscopic constant of the solvent (e.g., water \(K_f\approx 1.86\ \mathrm{°C\,kg\,mol^{-1}}\); benzene \(K_f\approx 5.12\ \mathrm{°C\,kg\,mol^{-1}}\)).
- \(m\) is the molality of the solute in the solution (mol of solute per kg of solvent).
How molality is built
Molality uses solute moles and solvent mass in kilograms:
\[
m \;=\; \frac{n_{\text{solute}}}{m_{\text{solvent}}[\mathrm{kg}]},\qquad
n_{\text{solute}} \;=\; \frac{m_{\text{solute}}}{M_r}
\]
Because molality references solvent mass (not volume), it is insensitive to temperature-dependent density changes and is preferred for colligative properties.
What the calculator can solve
- Forward: Given \(i\), \(K_f\), and either a direct \(m\) or masses (\(m_{\text{solute}},\,M_r,\,m_{\text{solvent}}\)), compute \(\Delta T_f\) and (optionally) \(T_f = T_f^{\circ} - \Delta T_f\).
- Inverse (cryoscopy): From a measured \(\Delta T_f\), find:
- \(m = \dfrac{\Delta T_f}{iK_f}\)
- \(M_r = \dfrac{i\,K_f\,m_{\text{solute}}}{\Delta T_f\,m_{\text{solvent}}[\mathrm{kg}]}\)
- \(m_{\text{solute}} = \dfrac{\Delta T_f\,m_{\text{solvent}}[\mathrm{kg}]\,M_r}{i\,K_f}\)
- \(m_{\text{solvent}}[\mathrm{kg}] = \dfrac{i\,K_f\,m_{\text{solute}}}{\Delta T_f\,M_r}\)
- \(i = \dfrac{\Delta T_f}{K_f\,m}\) (direct molality) or \(i = \dfrac{\Delta T_f\,M_r\,m_{\text{solvent}}[\mathrm{kg}]}{K_f\,m_{\text{solute}}}\) (mass path)
Units & consistency
- \(\Delta T_f\) may be entered or displayed in °C or K (same magnitude). Ensure \(T_f^{\circ}\) and \(T_f\) are expressed in the same unit for subtraction.
- \(K_f\) must match the displayed temperature unit: \(\mathrm{°C\,kg\,mol^{-1}}\) or \(\mathrm{K\,kg\,mol^{-1}}\).
- Molality \(m\) is always \(\mathrm{mol\,kg^{-1}}\).
Interpreting the van ’t Hoff factor \(i\)
For strong electrolytes dissociating to \(\nu\) ions, ideal \(i\approx \nu\). Real solutions often have \(i\lt \nu\) due to ion pairing or non-ideality, especially at higher concentrations.
For associating solutes (e.g., dimerization), \(i\) can be \(<1\).
Assumptions & limitations
- Solution is sufficiently dilute that ideal colligative behavior applies.
- Solute is non-volatile and does not itself crystallize at the solvent’s freezing point.
- \(K_f\) is taken at the working temperature and pressure; values are solvent-specific.
- Electrolyte dissociation may be incomplete; use an empirically determined \(i\) if available.
Worked mini-example
Example: \(10.0\ \mathrm{g}\) NaCl (\(M_r=58.44\ \mathrm{g\,mol^{-1}}\), take \(i=1.9\)) in \(100.0\ \mathrm{g}\) \(\mathrm{H_2O}\) (\(K_f=1.86\ \mathrm{°C\,kg\,mol^{-1}}\)). Find \(\Delta T_f\) and \(T_f\) (with \(T_f^{\circ}=0.00\ ^\circ\mathrm{C}\)).
\[
n=\frac{10.0}{58.44}=0.171\ \mathrm{mol},\qquad m_{\text{solv}}=0.100\ \mathrm{kg},\qquad
m=\frac{0.171}{0.100}=1.71\ \mathrm{mol\,kg^{-1}}
\]
\[
\Delta T_f = iK_f m = (1.9)(1.86\ \mathrm{°C\,kg\,mol^{-1}})(1.71\ \mathrm{mol\,kg^{-1}})=6.04\ ^\circ\mathrm{C}
\]
\[
T_f = T_f^{\circ}-\Delta T_f = 0.00 - 6.04 = -6.04\ ^\circ\mathrm{C}
\]
Tips for reliable results
- Use accurate solvent mass; small errors propagate directly into molality and \(\Delta T_f\).
- If experimental \(\Delta T_f\) is used to extract \(M_r\) or \(i\), keep solutions very dilute and account for known association/dissociation.
- Remember: \(\Delta T_f\) magnitude is the same in K and °C; only absolute temperatures require unit conversion.
