Theoretical Prediction of Ultrasonic Velocity in Binary Liquid Mixtures

Theoretical values of ultrasonic velocity in the binary mixtures of chlorobenzene with benzene have been evaluated at different temperatures from 303.15 K-323.15 K using Nomoto’s relation, ideal mixture relation, Junjie’s method, collision factor theory and free length theory. Theoretical values were compared with the experimental values and the validity of the theories was checked by applying the chi-square test for goodness of fit and by calculating the average percentage error (APE).


Introduction
Measurement of ultrasonic velocity gives the valuable information about the physicochemical behavior of the liquid and liquid mixtures.Several relations, semi-empirical formula and theories are available for the theoretical computation of ultrasonic velocity in liquid and liquid mixtures [1][2][3][4][5] .Further, the best suitable theory for the given molecular system under study is also picked out by calculating the average percentage error and chi-square test.

Experimental
The ultrasonic velocity was measured at different temperatures from 303.15 K-323.15K using a single crystal interferometer with a high degree of accuracy operating at a frequency of 2 MHz.The density was measured at different temperatures from 303.15 K-323.15K using specific gravity bottle method by the standard procedure.

Theory
The following are relations/theories used for the prediction of ultrasonic velocity in the binary liquid mixtures.

Results and Discussion
The experimental values along with the values calculated theoretically using Nomoto's relation, collision factor theory, free length theory, ideal mixture relation and Junjie's method for the system chlorobenzene + benzene at different temperatures 303.15 K-323.15K are given in Table 1.The validity of the theories was checked by applying Chi-square test and by calculating average percentage error.

Chi-square test for goodness of fit
According to Karl Pearson 6 , the Chi-square value is calculated using the formula, For (n-1) degrees of freedom, where, n is the number of data used.

Average percentage error (APE)
The average percentage Error 7 is calculated using the relation, Where, n-number of data used.Umix(obs) = experimental values of ultrasonic velocities.Umix(cal) = computed values of ultrasonic velocities.
It can be seen from Table 1 that the theoretical values of ultrasonic velocity computed by various theories show deviation from experimental values.The reason may be the limitations and approximations incorporated in these theories.1 shows that the Chi-square value and APE value are minimum for Nomoto's relation than those obtained by other theories.However for the binary mixtures of chlorobenzene + benzene these values are minimum for JM.When two liquids are mixed, the interaction between the molecules of the two liquids takes place because of the presence of various forces like dispersive force, charge transfer, hydrogen bonding dipole-dipole and dipole-induced dipole interactions.Hence the observed deviation shows that the molecular interaction is taking place between the unlike molecules in the liquid mixture 8 .Similar kinds of results were obtained by earlier workers [9][10][11] .

Conclusion
Ultrasonic velocities predicted using NOM, IMR, JM, CFT and FLT were compared with experimentally measured velocity values at different temperatures from 303.15 K-323.15K for the binary mixture of chlorobenzene+benzene.It may be concluded that nomoto's relation is best suited for the binary mixtures of chlorobenzene+benzene at all the temperatures.The observed deviation of theoretical values of velocity from the experimental values is attributed to the presence of intermolecular interactions in the system studied.
s constant, Temperature dependent constant, U ∞ = 1600 m/sec, N is the Avagadro's number.Free length theory (FLT) 2, represents the first and second component of the liquid mixture and the other symbols have their usual meanings.

Table 1 .
Experimental and computed values of Ultrasonic velocity for chlorobenzene + benzene system at 303.15 K-323.15K