Kinetic Theory
THERMAL EXPANSION
Experiments show that most of bodies increase their volume upon heating. The extent of expansion of various bodies is characterized by the temperature coefficient of expansion, or simply the coefficient of expansion. While considering solid which retain their shape during temperature variations, the distinction is made between (a) a change in their linear dimensions (viz. the dimensions in a certain direction), i.e. linear expansion, and (b) a change in the volume of a body, i.e. cubic expansion.
The coefficient of linear expansion is the quantity
where lo is the initial length at 0oC and lt is the length at a temperature t. From this expansion, we can find
The dimensions of
The coefficient of cubic expansion is the quantity
where Vo is the volume of a body at 0oC and Vt is its volume at a temperature t. From this equation, we obtain
The quantity
The coefficient of cubic expansion is about three times larger than the coefficient of linear expansion:
The coefficient of cubic expansion
What obeys the general laws of thermal expansion only at a temperature above 4 oC. From 0 oC to 4 oC, water contracts rather than expands. At 4 oC, water occupies the smallest volume, i.e. it has the highest density. At the bottom of deep lakes, there is denser water in winter, which remains the temperature of 4 oC even after the upper layer has been frozen.
Example 1
The lengths l1i = 100 m of iron wire and l1c = 100 m of copper wire are marked off at t1 = 20 oC. What is the difference in lengths of the wires at t2 = 60 oC? The coefficients of linear expansion for iron and copper are
Solution:
The elongation of the iron wires is
Substituting
Subtracting (1) from (2) and considering that
For low values of temperature t, when
Consequently, since
It can been seen that the deviation from a more exact value of 19.9 mm amounts to 0.1 mm, i.e. the relative error
Example 2
A solid body floats in a liquid at a temperature t = 0o C and is completely submerged in it at 50o C. What fraction d of volume of the body is submerged in the liquid at 0o C if
Solution:
In both the cases the weight of the body will be balanced by the force of buoyancy on it.
At to = 0 oC, the buoyancy is
where Vo is the volume of the body and
Equating the right-hand sides of equation (1) and (2), we get
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