Entropia de mistura

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Entropy of mixing

CY101 T. Pradeep 2007


The meaning of entropy: Entropy of mixing
Consider that a number of ideal gases are separated which
are present in a vessel. Let ni be the number of moles
of each gas and Vi is the volume it occupies. The total entropy,
S1 = ∑ni (Cv ln T + R ln Vi + Si) 1
Note the term Si. It is because,
dS = Cv(dT/T) + R(dV/V)
S = Cv ln T + R ln V + SoLet the partitions between the gases are removed and they are
allowed to mix. Let the volume the gases occupy is V. The entropy,
S2 = ∑ni (Cv lnCY101 T. Pradeep+ Si) 2
T + R ln V 2007


Assume that the pressure has not changed and there is no change in
Ratio of volume,
Vi/V = ni/n = Xi - the mole fraction.
Where n is the total number of moles.
Substituting Vi = VXi inEqn.1
S1 = ∑ni (Cv ln T + R ln V + R ln Xi + Si)
The increase in entropy, the entropy of mixing,
S2 – S1 = –∑ ni R ln Xi
Entropy of mixing of 1 mole of the ideal gas,
∆Sm = – R ∑ni/n ln Xi = –R ∑ Xi ln Xi
The fraction Xi is less than unity in all cases, the logarithm is
negative and thus ∆Sm is always positive.
Thus mixing of gases (eg. by diffusion), always results in
increase in entropy.Mixing is spontaneous!
This is in general true CY101 T. Pradeep 2007 liquid or solid.
of any material,


Entropy and disorder
Spontaneous processes net increase in entropy
randomness of distribution (mixing).

increase in

The diffusion of initially separated gases
result in an increase in entropy. The process has
increased the random distribution of molecules. Spontaneousconduction of heat results in the random distribution of kinetic energy
of the atoms. Thus spontaneous processes increase randomness,
at the same time increases entropy. Therefore, it is appropriate to
suggest a relationship between
entropy and randomness.
CY101 T. Pradeep 2007


This definition of entropy, that it is a measure of randomness, is
one of great value. Apart form thequantitative relationship, the
concept is of great value in understanding chemical processes
quantitatively. A measure of entropy changes gives an indication of
structural changes. The process of fusion involves increase in
disorder and therefore, the entropy increase. Greater the disorder,
greater the entropy increase. Therefore, the heat of fusion of ice
and benzene are 5.26 and 8.27 cal/deg/mol.The difference is
because in water there is considerable amount of order due to
hydrogen bonding while there is no such interaction in benzene.

CY101 T. Pradeep 2007


Entropy and probability
A correlation is possible between entropy and probability.
Suppose there are two bulbs, connected through a valve, and assume
that one bulb is evacuated and the other is full of gas. As thestopcock
is opened, according to second law, the gas will distribute uniformly.
We can explain this using the theory of probability.
Assume that there is just one molecule in the system. Here, the
probability that it can be found in one bulb is ½ , ie. one chance in two.
If we have 2 molecules, the probability that two will be found in one bulb is ¼
ie. (½)2. If there are N molecules, theprobability that all the
molecules will be found in one bulb is (½)N. For a bulb having
one litre volume and the pressure is one atmosphere, the number of
molecules is 1022.
CY101 T. Pradeep 2007


Even for 10-6 atm, the number of molecules is 1016.
Thus the probability for the molecules to reside in one bulb is very
small. This is also the probability for the molecules to return to onebulb after being distributed uniformly.
Thus it can be considered that the spontaneous process in which
the gas is distributed uniformly in two bulbs is associated with the
large probability. All spontaneous processes represent changes
from a less portable state to a more portable state.
Since processes are accompanied by increase in entropy it is
possible that there is a relation between...
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