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Most substances expand when the temperature increases. One important exception is water between 0°C and 4°C.
The expansion is due to the increased amplitude of vibration of the particles.
Consider a metal rod of original length
is the increase in length due to a temperature change, T. Experiments show that is directly proportional to T.
Now consider a rod of original length 2o (made of the same metal). Along the length of this rod there will be twice as many "layers" of atoms all increasing their amplitudes of vibration as the temperature increases. We would therefore expect this rod to expand by 2, for the same increase in temperature.
Experiments confirm this hypothesis.
We can therefore write
The constant is written as a and is called the linear expansivity of the metal.
Rearranging this result gives
where is the length when the temperature has changed by DT.