Difference between revisions of "Boiling-point elevation"

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'''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-[[electrolyte]]s, the difference in boiling point from that of the pure solvent is proportional to the [[Amount of substance|amount]] of solute.
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'''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-[[electrolyte]]s, the difference in boiling point from that of the pure solvent is proportional to the [[Amount of substance|amount]] of solute.<ref name="Atkins">{{Atkins4th|pages=167–69}}.</ref>
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'''Ebullioscopy''' is the experimental technique that uses boiling-point elevation to measure the [[molecular weight]] of compounds. '''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, although their development led to the formulation of [[Raoult's law]].
 
  
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'''Ebullioscopy'''<ref group="Note">The term "ebullioscopy" comes from the French ''ébullition'', boiling, which in turn is derived from the Latin ''ēbullīre'', to boil.</ref> is the experimental technique that uses boiling-point elevation to measure the [[molecular weight]] of compounds.<ref name="Atkins"/> '''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,<ref name="Atkins"/> although their development led to the formulation of [[Raoult's law]].
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<br clear=all/>
 
==Description==
 
==Description==
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The boiling-point elevation Δ''T''<sub>b</sub> is proportional to the [[molality]] of the solution ''m'': the proportionality constant is called the '''ebullioscopic constant''' ''K''<sub>eb</sub>.
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{| class="wikitable" align=right style="margin:0 0 0 0.5em;"
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:Δ''T''<sub>b</sub> = ''K''<sub>eb</sub>''m''
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|-
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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''. Hence, the molar mass of a solute can be calculated from the boiling-point elevation by
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! Solvent
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:''M'' = ''K''<sub>eb</sub>''w''/Δ''T''<sub>b</sub>
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! ''K''<sub>b</sub><br/>{{overline|K kg mol<sup>−1</sup>}}
 +
|-
 +
| [[Acetic acid]] || align=center | 3.07
 +
|-
 +
| [[Acetone]] || align=center | 1.71
 +
|-
 +
| [[Aniline]] || align=center | 3.22
 +
|-
 +
| [[Benzene]] || align=center | 2.53
 +
|-
 +
| [[Carbon disulfide]] || align=center | 2.37
 +
|-
 +
| [[Chloroform]] || align=center | 3.66
 +
|-
 +
| [[Cyclohexane]] || align=center | 2.79
 +
|-
 +
| [[Diethyl ether]] || align=center | 1.82
 +
|-
 +
| [[Nitrobenzene]] || align=center | 5.26
 +
|-
 +
| [[Tetrachloromethane]] || align=center | 4.95
 +
|-
 +
| [[Water]] || align=center | 0.51
 +
|-
 +
| colspan=2 | ''Data'': Kaye & Laby Tables of Physical<br/>& Chemical Constants<ref>{{Kaye&Laby | contribution = Crysocopic and ebullioscopic constants and enthalpies of fusion and of evaporation of some common solvents | url = http://www.kayelaby.npl.co.uk/chemistry/3_10/3_10_4.html | chapter = 3.10.4 | accessdate = 2011-03-27}}.</ref>
 +
|-
 +
|}
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The boiling-point elevation Δ''T''<sub>b</sub> is proportional to the [[molality]] of the solution ''m'': the proportionality constant is called the '''ebullioscopic constant''' ''K''<sub>b</sub>.
 +
:Δ''T''<sub>b</sub> = ''K''<sub>b</sub>''m''
 +
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''.<ref group="Note">In formal terms, ''w''&nbsp;= ''m''<sub>2</sub>/(''m''<sub>1</sub>+''m''<sub>2</sub>)&nbsp;≈ ''m''<sub>2</sub>/''m''<sub>1</sub> if ''m''<sub>2</sub>&nbsp;≪&nbsp;''m''<sub>1</sub> (here, ''m''<sub>2</sub> is the mass of solute and ''m''<sub>1</sub> is the mass of solvent).</ref> Hence, the molar mass of a solute can be calculated from the boiling-point elevation by
 +
:''M'' = ''K''<sub>b</sub>''w''/Δ''T''<sub>b</sub>
  
 
The same phenomenon can be described in terms of [[vapour pressure]] for any given temperature. If ''p''<sub>0</sub> is the vapour pressure of the pure solvent and ''p'' is the vapour pressure of the solution:
 
The same phenomenon can be described in terms of [[vapour pressure]] for any given temperature. If ''p''<sub>0</sub> is the vapour pressure of the pure solvent and ''p'' is the vapour pressure of the solution:
 
:(''p''<sub>0</sub> − ''p'')/''p''<sub>0</sub> = ''K''′<sub>R</sub>''m''
 
:(''p''<sub>0</sub> − ''p'')/''p''<sub>0</sub> = ''K''′<sub>R</sub>''m''
−
where ''K''′<sub>R</sub> 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 that a single constant (''K''<sub>R</sub>) 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:
+
where ''K''′<sub>R</sub> 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<ref>{{citation | last = Raoult | first = F.-M. | authorlink = François-Marie Raoult | title = Loi générale des tensions de vapeur des dissolvants | journal = C. R. Hebd. Seances Acad. Sci. | year = 1887 | volume = 104 | pages = 1430–33 | url = http://gallica.bnf.fr/ark:/12148/bpt6k30607.image.f1429.langEN}}. [http://web.lemoyne.edu/~giunta/raoult2.html English translation]</ref> that a single constant (''K''<sub>R</sub>) 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:<ref group="Note">In formal terms, ''x''&nbsp;= ''n''<sub>2</sub>/(''n''<sub>1</sub>+''n''<sub>2</sub>)&nbsp;≈ ''n''<sub>2</sub>/''n''<sub>1</sub> if ''n''<sub>2</sub>&nbsp;≪&nbsp;''n''<sub>1</sub> (here, ''n''<sub>2</sub> is the amount of solute and ''n''<sub>1</sub> is the amount of solvent).</ref>
 
:(''p''<sub>0</sub> − ''p'')/''p''<sub>0</sub> = ''K''<sub>R</sub>''x''
 
:(''p''<sub>0</sub> − ''p'')/''p''<sub>0</sub> = ''K''<sub>R</sub>''x''
 
where ''K''<sub>R</sub>&nbsp;≈&nbsp;1 (Raoult found a value of 1.05).
 
where ''K''<sub>R</sub>&nbsp;≈&nbsp;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. [[electrolyte]]s) or associates (e.g. [[acetic acid]] in non-polar solvents) in solution.
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 +
==Derivation==
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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
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:''μ''(ℓ) = ''μ''<sub>0</sub>(ℓ) + ''RT''ln(1−''x'')
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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.
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:''μ''<sub>0</sub>(g) = ''μ''(ℓ) = ''μ''<sub>0</sub>(ℓ) + ''RT''ln(1−''x'')
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:ln(1−''x'') = [''μ''<sub>0</sub>(g) − ''μ''<sub>0</sub>(ℓ)]/''RT'' = Δ<sub>vap</sub>''G''/''RT''
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From the definition of the [[Gibbs energy change of vaporization]] Δ<sub>vap</sub>''G''&nbsp;= Δ<sub>vap</sub>''H''&nbsp;−&nbsp;''T''Δ<sub>vap</sub>''S'', we can write two equations, the first for the solution and the second for the pure solvent (where ''T''<sub>0</sub> is the boiling point of the pure solvent):
 +
:ln(1−''x'') = Δ<sub>vap</sub>''H''/''RT'' − Δ<sub>vap</sub>''S''/''R''
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:ln(1) = Δ<sub>vap</sub>''H''/''RT''<sub>0</sub> − Δ<sub>vap</sub>''S''/''R''
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The difference of the two equations, remembering that ln(1)&nbsp;=&nbsp;0, is
 +
:ln(1−''x'') = (Δ<sub>vap</sub>''H''/''R'')[(1/''T'') − (1/''T''<sub>0</sub>)]
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Approximating ln(1−''x'')&nbsp;≈&nbsp;−''x'' (for ''x''&nbsp;≪&nbsp;1, i.e. dilute solution) gives
 +
:''x'' = (Δ<sub>vap</sub>''H''/''R'')[(1/''T''<sub>0</sub>) − (1/''T'')]
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As ''T''&nbsp;≈&nbsp;''T''<sub>0</sub> (small elevation of the boiling point), we can approximate (1/''T''<sub>0</sub>)&nbsp;−&nbsp;(1/''T'')&nbsp;≈ Δ''T''/''T''{{su|b=0|p=2}}, so
 +
:Δ''T'' = (''RT''{{su|b=0|p=2}}/Δ<sub>vap</sub>''H'')''x''
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The amount fraction of solute is equal to the molality of the solution ''m'' multiplied by the [[molar mass]] of the sovent ''M''<sub>0</sub>, so
 +
:Δ''T'' = (''RT''{{su|b=0|p=2}}''M''<sub>0</sub>/Δ<sub>vap</sub>''H'')''m''
 +
:''K''<sub>b</sub> = ''RT''{{su|b=0|p=2}}''M''<sub>0</sub>/Δ<sub>vap</sub>''H''
 +
 +
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 ''K''<sub>b</sub> are usually treated as empirical constants to be determined experimentally rather than being calculated from enthalpies of vaporization.<ref name="Atkins"/>
 +
 
==See also==
 
==See also==
 
*[[Freezing-point depression]]
 
*[[Freezing-point depression]]

Latest revision as of 08:04, 4 April 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[Note 1] 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 2] 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 3]

(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

  1. ↑ The term "ebullioscopy" comes from the French ébullition, boiling, which in turn is derived from the Latin ēbullīre, to boil.
  2. ↑ In formal terms, w = m2/(m1+m2) ≈ m2/m1 if m2 ≪ m1 (here, m2 is the mass of solute and m1 is the mass of solvent).
  3. ↑ In formal terms, x = n2/(n1+n2) ≈ n2/n1 if n2 ≪ n1 (here, n2 is the amount of solute and n1 is the amount of solvent).

References

  1. ↑ 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.
  2. ↑ 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).
  3. ↑ 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

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