Concise Physical Chemistry

c03 JWBS043-Rogers September 13, 2010 11:24 Printer Name: Yet to Come

40 THE THERMODYNAMICS OF SIMPLE SYSTEMS

We still use the practical method of length times cross-sectional area to get the total

volume which we then multiply byρ, but the corresponding integral is no longer a

conventional integral overdx; rather, it is the summed infinitesimal increments along

the length of the rodds, not coincident or even parallel to thexaxis.

Now suppose that the density of brass is not constant in the rod but that it varies

along the length of the rod according to some known functionρ(s). The integral is

M=

∫b

a

ρ(s)ds=ρ(s)(b−a)

Integrals taken along some curveCwith arc lengths, not one of the coordinate axes,

are calledcurvilinearorline integrals. In general, the integral of some function of

arc lengthf(s) along a curveCis

I=

∫

C

f(s)ds

3.3.1 Mathematical Interlude: The Length of an Arc

By Pythagoras’s theorem for the hypotenuse of a right triangle, the lengthsof a

short part of a curve (arc) inx–yspace is approximately (Barrante, 1998)

s≈

(

x^2 +y^2

) 1 / 2

Multiply and divide byxto find

s≈

(

12 +

(

y

x

) 2 ) 1 / 2

x

or, in the infinitesimal limit,

ds=

(

1 +

(

dy

dx

) 2 )^1 /^2

dx

For the length fromatobof afinitearc, find the integrals=

∫b

a

ds.

s

x

y

FIGURE 3.3 Pythagorean approximation to the short arc of a curve.