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Question: The volume of a metal sphere increases by \( 0.15\% \) when its temperature is raised by \( {24^ \ci...

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

Explanation

Solution

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 VV , the change in the volume of the metal sphere to be ΔV\Delta V and the change in the temperature be ΔT\Delta T .
Given, ΔV=0.15%\Delta V = 0.15\% of VV =0.0015V= 0.0015V
ΔT=24∘C\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, ΔV∝VΔT\Delta V \propto V\Delta T
By, removing the proportionality, we get
ΔV=VγΔT→\Delta V = V\gamma \Delta T \to equation (1)(1)
Here γ\gamma is the constant of proportionality which is called the coefficient of volume expansion
Substituting all the values in equation 11 , we get
0.0015V=Vγ×24 ⇒0.0015=γ×24 ⇒γ=6.25×10−5  0.0015V = V\gamma \times 24 \\\ \Rightarrow 0.0015 = \gamma \times 24 \\\ \Rightarrow \gamma = 6.25 \times {10^{ - 5}} \\\
We know that γ=3α\gamma = 3\alpha where α\alpha is the coefficient of linear expansion.
α=γ3\alpha = \dfrac{\gamma }{3}
substituting the value of γ\gamma in α\alpha , we get,
α=6.25×10−53 =2.0×10−5  \alpha = \dfrac{{6.25 \times {{10}^{ - 5}}}}{3} \\\ = 2.0 \times {10^{ - 5}} \\\
The coefficient of linear expansion of metal is α=2.0×10−5\alpha = 2.0 \times {10^{ - 5}} .
Therefore, the correct option is BB .

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.