Notes for Kinetic theory of Gases chapter of class 11 physics. Dronstudy provides free comprehensive chapterwise class 11 physics notes with proper images & diagram.
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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 equal to the fraction of the initial length by which a body taken at 0^{o}C has elongated as a result of heating it by 1^{o}C (or by 1 K):
where l_{Âo} is the initial length at 0^{o}C and l_{t} is the length at a temperature t. From this expansion, we can find
The dimensions of are K^{-1}Â (orÂ Â ^{o}C^{-1}).
The coefficient of cubic expansion is the quantity equal to the fraction of the initial volume by which the volume of a body taken at 0^{o}C has increased upon heating it by 1^{o}C (or by 1 K):
,
where V_{o} is the volume of a body at 0^{o}C and V_{t} is its volume at a temperature t. From this equation, we obtain
The quantity has also the dimensions of K^{-1} (orÂ ^{o}C^{-1}).
The coefficient of cubic expansion is about three times larger than the coefficient of linear expansion:
The coefficient of cubic expansion for liquids are somewhat higher than for solid bodies, ranging between 10^{-3} and 10^{-4} K^{-1}.
What obeys the general laws of thermal expansion only at a temperature above 4 ^{o}C. FromÂ 0 ^{o}C to 4 ^{o}C, water contracts rather than expands. At 4 ^{o}C, 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 ^{o}C even after the upper layer has been frozen.
Example 1
The lengths l_{1i} = 100 m of iron wire and l_{1c} = 100 m of copper wire are marked off at t_{1} = 20 ^{o}C. What is the difference in lengths of the wires at t_{2} = 60 ^{o}C? The coefficients of linear expansion for iron and copper are Â = 1.2 xÂ 10^{-5} K^{-1} and Â = 1.7 xÂ 10^{-5} K^{-1}.
Solution:
Â andÂ
The elongation of the iron wires is
.
Substituting Â , we find the elongation of the iron and copper wires
Â Â Â Â Â Â Â (1)
Â Â Â Â Â Â (2)
Subtracting (1) from (2) and considering that , we obtain
For low values of temperature t, when , it is not necessary to reduce l_{1} and l_{2} to l_{o1} and l_{Âo2} at t = 0 ^{o}C. To a sufficiently high degree of accuracy, we an assume that . Under this assumption, the problem can be solved in a simpler way:
Consequently, since , we have
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 = 0^{o }C and is completely submerged in it at 50^{o} C. What fraction d of volume of the body is submerged in the liquid at 0^{o} C ifÂ Â = 0.3 xÂ 10^{-5} K^{-2} and of the liquid, Â = 8.0 xÂ 10^{-5} K^{-1}?
Solution:
In both the cases the weight of the body will be balanced by the force of buoyancy on it.
Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â AtÂ t_{o} = 0 ^{o}C, the buoyancy is Â Â Â Â Â Â (1)
where V_{o} is the volume of the body and Â is the density of the liquid at t_{o} = 0^{o}C. At t = 50 ^{o}C, the volume of the body becomes Â and the density of the liquid is . The buoyancy in this case is
Â Â Â Â Â Â Â Â (2)
Equating the right-hand sides of equation (1) and (2), we get
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