Thermal Behavior of Matter and Heat Engines - Department of ...

Answer:
To make a washer from a disk of metal, we can cut along a circular curve, and remove
the inner disk. If we now heat the system, both the washer and the inner disk expand.
On the other hand, if we had left the inner disk in place and heated the original disk it
would also expand. Removing the heated inner disk would create an expanded
washer, with an expanded hole in the middle. We obtain the same result whether we
remove the inner disk and then heat, or heat first and then remove the inner disk.
Thus, heating the washer causes both it and its hole to expand, and they both expand
with the same coefficient of linear expansion.
11.13 Volume expansion
Similar to the argument in the expansion of area with temperature, the volume
changes with temperature. The initial volume of the cube is V = L 3 . If the temperature
of the cube is changed by ∆T, and the length of each side of the cube by ∆L,
3
3 3 3
2 3 2 3 3 3
V ' = ( L + ∆L)
= ( L + α L∆T
) = ( L + 3αL
∆T
+ 3α
L ∆T
+ α L ∆T
) .
2 3 2
3 3
Neglect the higher order term, e.g. α L ∆T and α L ∆T
3
, we have
3 3
V ' = L + 3αL
∆T
= V + 3αV∆T
That is
∆V
≈ 3 αV∆T
.
Though the calculation works out from cube, it is applicable to volume of any shape.
We define the coefficient of volume expansion, β where ∆V
≈ β V∆T
, and β = 3α.
Example
A copper flask with a volume of 150 cm 3 is filled to the brim with olive oil. If the
temperature of the system is increased from 6.0 o C to 31 o C, how much oil spills from
the flask? The coefficient of expansion of olive oil is given as 0.69 × 10 −3 K −1 .
Heat by 25 C o
4