Concise Physical Chemistry

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

36 THE THERMODYNAMICS OF SIMPLE SYSTEMS

inclined planes, one rough and the other smooth. The potential energy change is the

same, but the rough plane requires more work and it produces some amount ofheat

due to friction. Heat is not conserved over the cyclic path either. We symbolize the

nonconservation of work or heat by the sum, that is, the integral, of infinitesimal heat

or work incrementsdqordwover the cyclic path

∮

dw=0or

∮

dq=0. But we

have already said that energyUis conserved over a cyclic path so

∮

dU=0.

One of the great discoveries of Western science is that the infinitesimal increment

of the energy of a thermodynamic system is thesumof an infinitesimal increment in

work done on the system and an infinitesimal increment of the heat put into a system^1

dU=dw+dq

The statement

∮

dU=dq+dwis another of the many equivalent ways of stating the

first law of thermodynamics. Although the sum of workwand heatqis conserved,

we don’t know the ratio ofwtoqor even if they have the same sign, except by

experiment. The law of conservation of energy is the accumulated knowledge gained

from very many controlled observations over three centuries. It cannot be derived

from simpler principles.

3.1.1 The Reciprocity Relationship

For some functionsu=u(x,y), the differential

du=M(x,y)dx+N(x,y)dy

has the property that

M(x,y)=

(

∂u

∂x

)

y

and N(x,y)=

(

∂u

∂y

)

x

that is,

du=

(

∂u

∂x

)

y

dx+

(

∂u

∂y

)

x

dy

If this is true, thenduis called anexact differential. If it is not true, thenduis an

inexact differential. By the Euler reciprocity relationship, we have

∂^2 u

∂x∂y

=

∂^2 u

∂y∂x

[

∂

∂x

(

∂u

∂y

)

x

]

y

=

∂

∂x

N(x,y)=

[

∂

∂y

(

∂u

∂x

)

y

]

x

=

∂

∂y

M(x,y)

(^1) In the example of the mass being pushed up a rough plane, frictional heat is lost (goes out of the system);
hence the sign ondqis reversed:dU=dwin−dqout.