What is boiling point elevation?
Dissolving a non-volatile solute in a solvent raises the solvent’s boiling point.
For sufficiently dilute solutions that behave ideally, the elevation is proportional to the solution’s
molality, corrected for dissociation/association by the van ’t Hoff factor \(i\):
\[
\begin{aligned}
\Delta T_b &= i \cdot K_b \cdot m
\end{aligned}
\]
Here \(\Delta T_b\) is the temperature increase (same magnitude in K or °C),
\(K_b\) is the ebullioscopic constant of the pure solvent (units: K·kg·mol⁻¹ or °C·kg·mol⁻¹),
\(m\) is the solute molality (mol·kg⁻¹), and \(i\) counts the effective number of particles produced
per formula unit (e.g., for NaCl in water, \(i\approx2\); for glucose, \(i=1\)).
From \(\Delta T_b\) to the solution’s boiling point
The final boiling point of the solution adds the elevation to the pure solvent’s boiling point:
\[
\begin{aligned}
T_b &= T_b^{\circ} + \Delta T_b
\end{aligned}
\]
Where does \(K_b\) come from? (derivation sketch)
Starting from Raoult’s law and the Clausius–Clapeyron relation, for a dilute, ideal solution with a
non-volatile solute one obtains:
\[
\begin{aligned}
\Delta T_b &\approx \frac{R\,{T_b^{\circ}}^{2}}{\Delta H_{\mathrm{vap}}}\,x_{\text{solute}}
\end{aligned}
\]
For dilute solutions the solute mole fraction satisfies \(x_{\text{solute}} \approx m\,M_{\text{solv}}\), where \(M_{\text{solv}}\) is the molar mass of the solvent in kg·mol⁻¹. Substituting gives:
\[
\begin{aligned}
\Delta T_b &\approx \frac{R\,{T_b^{\circ}}^{2}}{\Delta H_{\mathrm{vap}}}\,\big(m\,M_{\text{solv}}\big)
\;=\; \Big(\frac{R\,{T_b^{\circ}}^{2}\,M_{\text{solv}}}{\Delta H_{\mathrm{vap}}}\Big)\,m \\[6pt]
\Rightarrow\quad
K_b &= \frac{R\,{T_b^{\circ}}^{2}\,M_{\text{solv}}}{\Delta H_{\mathrm{vap}}}
\end{aligned}
\]
with \(R\) in J·mol⁻¹·K⁻¹, \(\Delta H_{\mathrm{vap}}\) in J·mol⁻¹, and \(M_{\text{solv}}\) in kg·mol⁻¹,
yielding \(K_b\) in K·kg·mol⁻¹. In practice, you use tabulated \(K_b\) for each solvent.
Rearrangements used by the calculator
Solve for molality when \(\Delta T_b\) is measured:
\[
\begin{aligned}
m &= \frac{\Delta T_b}{\,i \cdot K_b\,}
\end{aligned}
\]
Build molality from masses (for a single solute): \(m = \dfrac{n_{\text{solute}}}{m_{\text{solv}}[\mathrm{kg}]} = \dfrac{m_{\text{solute}}/M_r}{m_{\text{solv}}[\mathrm{kg}]}\).
\[
\begin{aligned}
m &= \frac{\dfrac{m_{\text{solute}}}{M_r}}{m_{\text{solv}}[\mathrm{kg}]}
\end{aligned}
\]
Find molar mass from ebullioscopy (known \(\Delta T_b\), masses, \(K_b\), \(i\)):
\[
\begin{aligned}
\Delta T_b &= i\,K_b\,\frac{m_{\text{solute}}/M_r}{m_{\text{solv}}[\mathrm{kg}]}
\;\;\Longrightarrow\;\;
M_r \;=\; \frac{i\,K_b\,m_{\text{solute}}}{\Delta T_b \, m_{\text{solv}}[\mathrm{kg}]}
\end{aligned}
\]
Find van ’t Hoff factor from a measured elevation:
\[
\begin{aligned}
i &= \frac{\Delta T_b}{\,K_b \cdot m\,}
\end{aligned}
\]
Worked example (matches the “Fill example”)
Dissolve 10.0 g NaCl in 100.0 g H₂O. Use \(K_b(\text{water}) = 0.512\ \!^{\circ}\mathrm{C}\cdot\mathrm{kg}\cdot\mathrm{mol}^{-1}\),
\(M_r(\text{NaCl}) = 58.44\ \mathrm{g\cdot mol^{-1}}\), \(i \approx 1.9\), \(T_b^{\circ} = 100.00\ ^{\circ}\mathrm{C}\).
\[
\begin{aligned}
n_{\text{NaCl}} &= \frac{10.0\ \mathrm{g}}{58.44\ \mathrm{g\cdot mol^{-1}}}
&= 0.171\ \mathrm{mol} \\[6pt]
m_{\text{solv}}[\mathrm{kg}] &= 0.100\ \mathrm{kg} \\[6pt]
m &= \frac{0.171\ \mathrm{mol}}{0.100\ \mathrm{kg}}
&= 1.71\ \mathrm{mol\cdot kg^{-1}} \\[10pt]
\Delta T_b &= i\,K_b\,m
&= 1.9 \cdot 0.512 \cdot 1.71
&= 1.67\ ^{\circ}\mathrm{C} \\[6pt]
T_b &= 100.00 + 1.67
&= 101.67\ ^{\circ}\mathrm{C}
\end{aligned}
\]
Assumptions, units, and common pitfalls
- Dilute, ideal behavior. The relations assume low solute concentration and an ideal (Raoult-law) solvent.
- Non-volatile solute. The solute must not contribute appreciably to the vapor pressure.
- Units. Report \(K_b\) in K·kg·mol⁻¹ or °C·kg·mol⁻¹; \(\Delta T_b\) has the same magnitude in K or °C.
- van ’t Hoff factor \(i\). Use an effective \(i\) at the given concentration (electrolytes often have \(i\lt\) ideal due to ion pairing).
- Mass basis. Molality uses solvent mass only (kg), not total solution mass.
- Beyond ideality. At higher concentrations, activity coefficients are needed and the linear relation may break down.
