Question

The volume of a metal sphere increases by $ 0.15\% $ when its temperature is raised by $ {24^ \circ }C $ . The coefficient of linear expansion of metal is:
 $
  A)2.5 \times {10^{ - 5}}^ \circ {C^{ - 1}} \\
  B)2.0 \times {10^{ - 5}}^ \circ {C^{ - 1}} \\
  C) - 1.5 \times {10^{ - 5}}^ \circ {C^{ - 1}} \\
  D)1.2 \times {10^{ - 5}}^ \circ {C^{ - 1}} \\
  $

Answer 1

Hint: The term "expansion" refers to a change or increase in length. Linear expansion is defined as a change in length in one dimension (length) over a volume. The change in temperature is the cause of the expansion in this case. As a result, it is assumed that a change in temperature would result in a change in the rate of expansion.

Complete answer:
Assume the initial volume of the metal sphere to be $ V $ , the change in the volume of the metal sphere to be $ \Delta V $ and the change in the temperature be $ \Delta T $ .
Given, $ \Delta V = 0.15\% $ of $ V $ $ = 0.0015V $
 $ \Delta T = {24^ \circ }C $
As we already know, the change in volume of the metal sphere is proportional to the initial volume of the metal sphere and temperature change.
So, $ \Delta V \propto V\Delta T $
By, removing the proportionality, we get
 $ \Delta V = V\gamma \Delta T \to $ equation $ (1) $
Here $ \gamma $ is the constant of proportionality which is called the coefficient of volume expansion
Substituting all the values in equation $ 1 $ , we get
 $
  0.0015V = V\gamma \times 24 \\
   \Rightarrow 0.0015 = \gamma \times 24 \\
   \Rightarrow \gamma = 6.25 \times {10^{ - 5}} \\
  $
We know that $ \gamma = 3\alpha $ where $ \alpha $ is the coefficient of linear expansion.
 $ \alpha = \dfrac{\gamma }{3} $
substituting the value of $ \gamma $ in $ \alpha $ , we get,
 $
  \alpha = \dfrac{{6.25 \times {{10}^{ - 5}}}}{3} \\
   = 2.0 \times {10^{ - 5}} \\
  $
The coefficient of linear expansion of metal is $ \alpha = 2.0 \times {10^{ - 5}} $ .
Therefore, the correct option is $ B $ .

Note:
As the force of interaction between its molecules is weaker than that of solids, the liquid has no fixed shape. As a result, in the case of a liquid substance, linear and superficial expansion are meaningless. Its volume is proportional to the temperature. The volume of the liquid is unaffected by pressure.
The volume of the liquid expands in two distinct ways. There are two types of expansion: real and apparent.