Difference between revisions of "Boiling-point elevation"
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We assume [[ideal solution]] behaviour, so that the [[chemical potential]] of the solvent in the solution ''μ''(ℓ) is given by | We assume [[ideal solution]] behaviour, so that the [[chemical potential]] of the solvent in the solution ''μ''(ℓ) is given by | ||
:''μ''(ℓ) = ''μ''<sub>0</sub>(ℓ) + ''RT''ln(1−''x'') | :''μ''(ℓ) = ''μ''<sub>0</sub>(ℓ) + ''RT''ln(1−''x'') | ||
| − | where ''μ''<sub>0</sub>(ℓ) is the chemical potential of the pure solvent and ''x'' is the [[amount fraction]] of solute (so that 1−''x'' is the amount fraction of solvent). The solution | + | where ''μ''<sub>0</sub>(ℓ) is the chemical potential of the pure solvent and ''x'' is the [[amount fraction]] of solute (so that 1−''x'' is the amount fraction of solvent). The solution boils when the solvent in the solution is in equilibrium with pure solvent vapour above the solution, assuming that the solute is involatile, so ''μ''(ℓ) must be equal to ''μ''<sub>0</sub>(g), the chemical potential of the pure solvent vapour. |
:''μ''<sub>0</sub>(g) = ''μ''(ℓ) = ''μ''<sub>0</sub>(ℓ) + ''RT''ln(1−''x'') | :''μ''<sub>0</sub>(g) = ''μ''(ℓ) = ''μ''<sub>0</sub>(ℓ) + ''RT''ln(1−''x'') | ||
:ln(1−''x'') = [''μ''<sub>0</sub>(g) − ''μ''<sub>0</sub>(ℓ)]/''RT'' = Δ<sub>vap</sub>''G''/''RT'' | :ln(1−''x'') = [''μ''<sub>0</sub>(g) − ''μ''<sub>0</sub>(ℓ)]/''RT'' = Δ<sub>vap</sub>''G''/''RT'' | ||
Revision as of 13:04, 27 March 2011
Boiling-point elevation is the increase in temperature of the boiling point of a liquid that contains non-volatile solute. It is a colligative property: for dilute solutions of non-electrolytes, the difference in boiling point from that of the pure solvent is proportional to the amount of solute.[1]
Ebullioscopy is the experimental technique that uses boiling-point elevation to measure the molecular weight of compounds.[1] Tonometry is a related technique that directly measures the reduction in vapour pressure of the solution compared to that of the pure solvent. Both techniques are of historical interest only,[1] although their development led to the formulation of Raoult's law.
Description
| Solvent | Kb K kg mol−1 |
|---|---|
| Acetic acid | 3.07 |
| Acetone | 1.71 |
| Aniline | 3.22 |
| Benzene | 2.53 |
| Carbon disulfide | 2.37 |
| Chloroform | 3.66 |
| Cyclohexane | 2.79 |
| Diethyl ether | 1.82 |
| Nitrobenzene | 5.26 |
| Tetrachloromethane | 4.95 |
| Water | 0.51 |
| Data: Kaye & Laby Tables of Physical & Chemical Constants[2] | |
The boiling-point elevation ΔTb is proportional to the molality of the solution m: the proportionality constant is called the ebullioscopic constant Kb.
- ΔTb = Kbm
The molality of the solution is the amount of solute divided by the mass of solvent. For dilute solutions, where the mass of the solution can be approximated by the mass of solvent, the molality is equal to the mass fraction w of solute divided by its molar mass M.[Note 1] Hence, the molar mass of a solute can be calculated from the boiling-point elevation by
- M = Kbw/ΔTb
The same phenomenon can be described in terms of vapour pressure for any given temperature. If p0 is the vapour pressure of the pure solvent and p is the vapour pressure of the solution:
- (p0 − p)/p0 = K′Rm
where K′R is a constant that is specific for each solvent and m is the molality of the solution. French physical chemist François-Marie Raoult (1830–1901) investigated this relation in the 1880s, and found in 1887[3] that a single constant (KR) for all solvents could be obtained by replacing the molality with the amount fraction of solute x, that is by dividing the molality by the molar mass of the solvent:[Note 2]
- (p0 − p)/p0 = KRx
where KR ≈ 1 (Raoult found a value of 1.05).
Both of these descriptions only apply for dilute solutions. They also break down if the solute dissociates (e.g. electrolytes) or associates (e.g. acetic acid in non-polar solvents) in solution.
Derivation
We assume ideal solution behaviour, so that the chemical potential of the solvent in the solution μ(ℓ) is given by
- μ(ℓ) = μ0(ℓ) + RTln(1−x)
where μ0(ℓ) is the chemical potential of the pure solvent and x is the amount fraction of solute (so that 1−x is the amount fraction of solvent). The solution boils when the solvent in the solution is in equilibrium with pure solvent vapour above the solution, assuming that the solute is involatile, so μ(ℓ) must be equal to μ0(g), the chemical potential of the pure solvent vapour.
- μ0(g) = μ(ℓ) = μ0(ℓ) + RTln(1−x)
- ln(1−x) = [μ0(g) − μ0(ℓ)]/RT = ΔvapG/RT
From the definition of the Gibbs energy change of vaporization ΔvapG = ΔvapH − TΔvapS, we can write two equations, the first for the solution and the second for the pure solvent (where T0 is the boiling point of the pure solvent):
- ln(1−x) = ΔvapH/RT − ΔvapS/R
- ln(1) = ΔvapH/RT0 − ΔvapS/R
The difference of the two equations, remembering that ln(1) = 0, is
- ln(1−x) = (ΔvapH/R)[(1/T) − (1/T0)]
Approximating ln(1−x) ≈ −x (for x ≪ 1, i.e. dilute solution) gives
- x = (ΔvapH/R)[(1/T0) − (1/T)]
As T ≈ T0 (small elevation of the boiling point), we can approximate (1/T0) − (1/T) ≈ ΔT/T20, so
- ΔT = (RT20/ΔvapH)x
The amount fraction of solute is equal to the molality of the solution m multiplied by the molar mass of the sovent M0, so
- ΔT = (RT20M0/ΔvapH)m
- Kb = RT20M0/ΔvapH
This derivation shows why boiling-point elevation is proportional to the amount fraction of solute and hence, for dilute solutions, to the molality. Nevertheless, ebullioscopic constants Kb are usually treated as empirical constants to be determined experimentally rather than being calculated from enthalpies of vaporization.[1]
See also
Notes and references
Notes
References
- ↑ 1.0 1.1 1.2 1.3 Atkins, P. W. Physical Chemistry, 4th ed.; University Press: Oxford, 1990; pp 167–69. ISBN 0-19-855283-1.
- ↑ Crysocopic and ebullioscopic constants and enthalpies of fusion and of evaporation of some common solvents. In Kaye & Laby Tables of Physical & Chemical Constants, 16th ed., 1995; Chapter 3.10.4, <http://www.kayelaby.npl.co.uk/chemistry/3_10/3_10_4.html>. (accessed 27 March 2011).
- ↑ Raoult, F.-M. Loi générale des tensions de vapeur des dissolvants. C. R. Hebd. Seances Acad. Sci. 1887, 104, 1430–33, <http://gallica.bnf.fr/ark:/12148/bpt6k30607.image.f1429.langEN>. English translation
External links
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See also the corresponding article on Wikipedia. |
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