Boiling-point Elevation JOSEPH J O F F E Newark College of Engineering, Newark, New Jersey
I . .
N THIS paper is . derived . a law for boiling-point ele-
vatlon in solutions whlch obey Raoult's Law, which holds up to considerably higher concentrations than the usual forms of this law for dilute solutions.
BC which cuts the vapor pressure curve of the solution a t B and C:
-
A 4 = slope of BC
AT
(1)
The slope of the secant BC may be approximated by the slope of the tangent to the curve a t B, which is LP given by the Clapeyron-Clausius equation, d~ - = -. dT RT2 For a solution obeying Raoult's Law L is identical with the molal latent heat of vaporization of the solvent from the pure solvent.' Hence, equation (1) becomes
Consider the vapor pressure curves of solvent and solution shown in the figure. The solvent boils a t a temperature To and a pressure po. The solution attains this pressure a t temperature T. The vapor pressure lowering Ap = PO - p and the boiling-point elevation AT = T - To are related by the slope of the secant
But by Raoult's Law p, = NIP,and Ap = N%, where N , is the mol fraction of the solvent and Nz is the mol fraction of the solute. Hence, from equation ( Z ) , G~ass~oNe. S., "Text-Book of Physical Chemistry,'' D. Van Nostrand Company, Inc., New York, 1940, p. 616.
R T 2 N* A T = -RTo' - = - -Ap L NLPA L NL
(3)
or, since N ~ / N I= m/nl, where n2 is the number of mols of solute and nl is the number of mols of solvent, AT = RToa -.9 n
L
AT =
-
[&-I
- -.
RT'
- Lpc - L dT o RTa
-
Ap -
RTP AT = - N* L
(7)
-
L
ln(1- N*)
(8)
~ -ln ( 1 -~ N ~ in) an~infinite sefies3 ~ d we have: -In (1 - N*) =
N*+ 1 / 2 N 2+ 'laNa5+ . . .
n,
-
-Nz =
N,
1
-
N' -N,+NzP+Np3+... Na
+
%
nl
+ '/m
N2 in an infinite series: 1 - l/2N%
- L/2N, = AT, + 1/,Ns2 + l/.N2 Nz
RTTo AT=-. L n,
+ .. .
(12)
(lo)
so that the mol ratio can be substituted for -In (1 N2) giving equation ( 4 ) . A much better aDDroximation than either 'eanation ( 4 ) or ( 7 ) can he odtkned by constructing a tangent to the vapor pressure curve of the solution a t a point interNOYES,A. A,. AND M . S. SHERRILL, "A Course of Study in
+ L/m~ 12%
(13)
which is a close approximation of equation ( 8 ) . For greatest accuracy a mean value of L should be used, corresponding to the mean of temperatures To and T. the table below are given values of -ln(l - N ~ ) , np/nl,and of n l / ( n l + I / ~ which ~ ~ ) correspond to cer. tain values of N ~ . - 1 (I
-N
NI
n,/m
+ Vmr
n.
0.01005
0 . 0 2 0 2 0 0.05129
O.OIOOO 0 . 0 2 0 0 0
I2
0.1054
0.2231
0.6932
0 . 0 5 0 ~ o.1000
0.2000
O.SM)O
0.01010
0.02041
0.05263
O . I I I I 0.2500
I.OM)O
0.01005
0.02020
0.05128
0.1053
0.6667
0.2222
It is seen from inspection of the table that a t a mol fraction of ' / I D the error in usinc the mol fraction or the rnoli ratio in place of the~ logariihmic function is 5.5 per ~ cent, while the error resulting from the expression
(9)
Obviously, equation ( 7 ) follows from (8) if in the expansion all terms in N2 after the first are neglected. On the other hand,&
3
RTD' nn L ' n, L/*nn (11)
act" boiling-point law is shown by expanding
(6)
The law of boiling-point elevation is usually derived in this form and equation ( 4 ) is obtained from it by a second approximation, since in dilute solutions N z Equation ( 7 ) gives a value for AT which is nz/nl. too small, since the slope of the tangent a t Cis steeper than that of the secant BC. In fact, equations ( 4 ) and ( 7 ) give deviations in opposite directions from the "exact" law of boiling-point elevation which is obtained by integration of the Clapeyron-Clausius equation. The "exact" law is2
-
N*
1- '/*N* -
That this result is a better approximation of the "ex-
To
AT =
AP - RTD' - - AP L ' P A - I / ~ A P-
We see that the first two terms of this series are identical with those of the series in (9) and that the third term differs by only '/12N2a.If it is assumed that T , is the geometric mean of To and T,equation ( 1 1 ) becomes
PC
RTa AT = 2N, L
~
-
RTD' L
I
With the aid of Raoult's Law this becomes
or, since T
APAT = --
(4)
nl
Equation ( 4 ) is the form in which the law of boilingpoint elevation in dilute solutions is usually given. Since the slope of the secant BC is steeper than that of the tangent a t B , i t can he seen from the figure that the value for AT obtained from equation ( 4 ) is too large. On the other hand, if the slope of secant BC is approximated by that of the tangent to the curve a t C, Ap
mediate between B and C. Thus, if point D is chosen a t which p, = p, - ' / Z A P we , have:
722 nl
$.
tion
'/,%
is about 0.1 per cent. Even a t a mol fracthe expression
+
involves an n, 'Inn2 error of only 3.8 per cent. It may he concluded that equation (13) may be used in placeof equation (8) even in fairly concentrated solutions which obey Raoult's Law. For the purpose of calculating molecular weights of dissolved solutes equation (4) is often written in the form: N 2
= '/z
1000u,Kb =
YiZr
(14)
Chemical Principles," 2nd Edition, The Macmillan Company, where M2 is the molecular weight of the solute, wl and New York, 1938, pp. 227-9. JSee, for example. EUCKEN, A., E. R . JETTE, ANDV. LAMER. w2 are the weights of solvent and solute, respectively, "Fundamentals of Physical Chemistry," McGraw-Hill B w k and KBis the hoiling-point constant, Company, Inc., New York, 1925, p. 21. Similarly, starting with (13) we have: Ibid., p. 213.
Consequently:
It can also be shown that if p~ is assumed to be the geometric rather than the arithmetic mean of p, and PB, a still better algebraic approximation to equation (8) is obtained:
for which the expansion is:
The second term on the right of equation (18) may be looked upon as a correction term which can be applied to the value of the molecular weight as usually obtained from (14). Thus, for a molal solution of naphthalene in benzene (Mi = 128) the correction amounts to -0.40 or 0.3 per cent.
Equation (19), however, is not as well adapted to numerical computation as (13), particularly when i t is required to solve for the molecular weight of the solute. The above analysis can be extended to freezing-point lowering, yielding equations for AT and Mz exactly analogous to equations (13), (IS), and (19).
