Cálculo da Composição e Volume

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SOLUTIONSMANUAL TO ACCOMPANY ATKINS' PHYSICAL CHEMISTRY 119
�e composition of the vapour – that is the mole fractions of A and B in the
vapour – is calculated from their partial pressures according to [1A.6–9], pA =
yAp, where yA is themole fraction of A in the vapour and p is the total pressure.
�e partial pressure of A is also given by pA = xAp∗A.�ese two expressions for
pA is equated to give
yAp = xAp∗A hence yA =
xAp∗A
p
�e composition of the vapour is therefore
yA =
xAp∗A
p
=
1
2 × (20 kPa)
19 kPa
= 0.53
yB =
xBp∗B
p
=
1
2 × (18 kPa)
19 kPa
= 0.47
E5A.6(b) �e total volume is calculated from the partial molar volumes of the two com-
ponents using [5A.3–144], V = nAVA + nBVB.�e task is therefore to �nd the
amount in moles, nA and nB, of A and B in a given mass m of solution. If the
molar masses of A and B areMA andMB then it follows that
m = nAMA + nBMB
�e mole fraction of A is de�ned as xA = nA/(nA + nB), hence nA = xA(nA +
nB) and likewise for B. With these substitutions for nA and nB the previous
equation becomes
m = xAMA(nA + nB) + xBMB(nA + nB) hence (nA + nB) =
m
xAMA + xBMB
�is latter expression for the total amount in moles, (nA + nB), is used with
nA = xA(nA + nB) to give
nA = xA(nA + nB) =
mxA
xAMA + xBMB
and likewise
nB =
mxB
xAMA + xBMB
With these expressions for nA and nB the total volume is computed from the
partial molar volumes
V = nAVA + nBVB =
mxAVA
xAMA + xBMB
+ mxBVB
xAMA + xBMB
= m
xAMA + xBMB
[xAVA + xBVB]
= m
xAMA + (1 − xA)MB
[xAVA + (1 − xA)VB]
where on the last line xB = (1 − xA) is used.