P19_007.PDF

7. (a) Changesintemperaturetakeplacebymeansofradiation, conduction, andconvection. Theconstant

A can be reduced by placing the object in isolation, by surrounding it with a vacuum jacket, for

example. This reduces conduction and convection. Absorption of radiation can be reduced by

polishing the surface to a mirror ﬁnish. We note that A depends on the condition of the surface

and on the ability of the environment to conduct or convect energy to or from the object. A has

the dimensions of reciprocal time.

(b) We rearrange the equation to obtain

∆ T

d ∆ T

−

Now, we integrate with respect to time and recognize that

∆ T

d ∆ T

dt = 1

∆ T d (∆ T ) .

Thus,

∆ T

∆ T d (∆ T )=

−

Adt .

∆ T 0

The integral on the right side yields

−

At and the integral on the left yields ln∆ T

∆ ∆ T 0 =ln(∆ T )

−

ln(∆ T 0 )=ln(∆ T/ ∆ T 0 ), so

ln ∆ T

∆ T 0 =

−

At .

We use each side as the exponent of e , the base of the natural logarithms, to obtain

∆ T

∆ T 0 = e − At

∆ T =∆ T 0 e − At