PHYSICAL CHEMISTRY IN BRIEF

CHAP. 3: FUNDAMENTALS OF THERMODYNAMICS [CONTENTS] 81

3.3.3 Total differential of the product and ratio of two functions

Similar rules as those governing differentiation hold for the total differential of the product and

the ratio of functionsxandy.

d(xy) =xdy+ydx , d

(

x

y

)

=

1

y

dx−

x

y^2

dy. (3.29)

For example, according to (3.29) we have for the total differential of the product of volume and

pressure

d(pV) =pdV+Vdp.

3.3.4 Integration of the total differential

The integral of the total differential from point (x 1 ,y 1 ) to point (x 2 ,y 2 ) does not depend on the

path between these points. We can, e.g., first integrate with respect toxat a fixedy 1 , and

then with respect toyat a fixedx 2

z(x 2 , y 2 ) =z(x 1 , y 1 ) +

∫x 2

x 1

(

∂z

∂x

)

y=y 1

dx+

∫y 2

y 1

(

∂z

∂y

)

x=x 2

dy , (3.30)

or we can first integrate with respect toyat a fixedx 1 and then with respect toxat a fixedy 2

z(x 2 , y 2 ) =z(x 1 , y 1 ) +

∫y 2

y 1

(

∂z

∂y

)

x=x 1

dy+

∫x 2

x 1

(

∂z

∂x

)

y=y 2

dx , (3.31)

or we can choose any other path linking points (x 1 ,y 1 ) and (x 2 ,y 2 ) in the planex, y.

S Symbols:The expression

(

∂z

∂x

)

y=y 1

denotes the partial derivative of the functionz=f(x, y)

with respect toxat constantyat the valuey=y 1.