Kinetic energies of gas molecules - ACS Publications

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GUEST AUTHOR John C. Aherne

Textbook Errors, 63

Crawford Municipal Technical Institute Cork, Ireland

Kinetic Energies of GUS Molecules

T h e distribution of kinetic energy among molecules of a gas a t a given temperature is often discussed in connection with the study of the gas laws or of reaction rates. I n the more elementary treatment this discussion is usually qualitative but is commonly illustrated by a graph showing a plot of "fraction of molecules" versus "energy." In some textbooks' the curve representing this relation has the general characteristics of the curve shown in Figure 1, i.e., it has points of inflection on both sides of the maximum. This is incorrect. Figure 1 actually represents the distribution of molecular speeds. The

That this is so follows in a straightforward manner from the Maxwell-Boltzmann relation. According to this law, the fraction of molecules ( l / n O ) d n having

Figure 2.

Distribution of kinetic energies.

speeds c (i.e. velocities irrespective of direction) between c and c dc is given by2

+

This relation is more readily manipulated in its reduced form, i.e., when the speeds are expressed in terms of the modal velocity ( 2 k T / m ) ' / e . We therefore substitute c

=

c(2kT/m)'/x

and

de = (2kT/m)'/rdc

Equation (I) then simplifies to "

Figure 1.

c. Speed /iZhT/m Distribution of motocular sped.

I"*

proper shape of the curve for the distribution of kinetic energies is shown in Figure 2 . It has a point of inflection only on the high energy side of the maximum. Suggestions of material suitable for this column and guest columns suitable for publication directly should be sent with as many details ss possible, and pertioulsrly with references to modern textbooks, to W. H. Eherhardt, Department of Chemistry, Georgia Institute of Technology, Atlanta, Georgia 30332. Since the purpose of this column is to prevent the spread and continuation of errors and not the evaluation of individual texts, the sources of errors discussed will not be cited. I n order to he presented an error must ocour in at leaat two independent recent standard books.

I n order to find the corresponding relation for the energy distribution we only need to realize that E/kT = 1/Bmc2/kT= c3 = E

(3)

where E is the kinetic energy expressed in units of k T . Now d E = 2c dc and hence from equation ( 2 ) we find

A plot of the left hand side of equation (4) against E 2

A method which is relativelv ~ainlessto the bepinner for

makes use of histograms and of the reduced form of the equation.

Volume 42, Number 12, December 1965

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655

(or if one wishes to illustrate the effect of temperature, against E ) gives the desired graph (Fig. 2). We may now locate the extrema and points of inflection of the curve in the usual manner by diierentiating twice with respect to E and by determining the roots of these derivatives. We find

physical reality. The curve approaches the origin tangentially to the ordinate. On the other hand successive differentiation of equation (2) with respect to c yields

=

0 for E = I/, = mfarE=O =

=

0 for E

=

1.207 or -0.207

Thus the curve in Figure 2 has a maximum at E = (or E = '/zkT) and one point of inflection at E = 1.207 (or 1.207kT). However, the second point of inflection occurs a t a negative value of E and hence has no

656 / Journol of Cherniccd Education

=

+I, 0

and

and

=

0 for c

0 for c

=

&1.51, &0.47

Hence the curve has a maximum a t the modal speed ( M V ) ,has two points of inflection (one a t 0.47 M V and the other a t 1.51 M V ) , and approaches the origin tangentially to the c axis. Note that the roots for negative values of c have no physical significance, since we are here considering speeds, i.e., velocities without directions.