Question

A meter rod of silver at 0°C is heated to 100°C. Its length is increased by 0.19 cm. Coefficient of volume expansion of the silver rod is:
${\text{A}}{\text{. 5}}{\text{.7 }} \times {\text{ 1}}{{\text{0}}^{ - 5}}/^\circ {\text{C}} \\\ {\text{B}}{\text{. 0}}{\text{.63 }} \times {\text{ 1}}{{\text{0}}^{ - 5}}/^\circ {\text{C}} \\\ {\text{C}}{\text{. 1}}{\text{.9 }} \times {\text{ 1}}{{\text{0}}^{ - 5}}/^\circ {\text{C}} \\\ {\text{D}}{\text{. 16}}{\text{.1 }} \times {\text{ 1}}{{\text{0}}^{ - 5}}/^\circ {\text{C}} \\\$

Answer 1

Hint – To find the coefficient of volume expansion, we apply the formula of coefficient of linear expansion, use the given data and find its value. Using this value we compute the coefficient of volume expansion.

Formula Used:
Coefficient of linear expansion $\alpha {\text{ = }}\dfrac{{\Delta {\text{l}}}}{{{{\text{l}}_{{\text{initial}}}} \times \Delta {\text{T}}}}$
Where ∆l is the change in length, ${{\text{l}}_{{\text{initial}}}}$is the initial length of the rod before expansion and ∆T is the change in temperature.

Complete step-by-step solution:
Given Data,
Change in length, i.e. $\Delta {\text{L}}$ = 0.19 cm
Change in temperature = $\Delta {\text{T}}$ = 100 – 0 = 100°C
Let us consider the initial length to be 100 cm, ${{\text{l}}_{{\text{initial}}}}$= 100 cm.
We know the volume of a solid shape of some material is given by the product of its length, breadth, and height. Volume is a 3-dimensional attribute of a body essentially governed by its length in each of the axes.
According to the given data, we know the formula volume of the silver rod initially is, $V = πr^{2} \times {{\text{l}}_{{\text{initial}}}}$
Hence, the change in volume in the rod is governed by the change in length, we establish a relation between these two.
Where r and l are the radii of the base and initial length of the rod. As the radius term is not mentioned we consider the radius of the rod remains constant throughout the process.
The ratio increase in length original length for 1 degree rise in temperature is called the coefficient of linear expansion and it is given by:
$\alpha {\text{ = }}\dfrac{{\Delta {\text{l}}}}{{{{\text{l}}_{{\text{initial}}}} \times \Delta {\text{T}}}}$
$\Rightarrow \alpha {\text{ = }}\dfrac{{0.19}}{{{\text{100 }} \times {\text{ 100}}}} \\\ \Rightarrow \alpha {\text{ = 1}}{\text{.9 }} \times {\text{1}}{{\text{0}}^{ - 5}}/^\circ {\text{C}}$
Volume expansion is defined as the increase in the volume of the solid on heating. With a change in temperature ∆T the change in volume of a solid is given by $∆V = {{\text{V}}_{\text{y}}} \times \Delta{\text{t}}$, where the coefficient of volume expansion is y.
And y is given as ${\text{y = }}\dfrac{{\Delta {\text{V}}}}{{{{\text{V}}_{{\text{initial}}}} \times \Delta {\text{T}}}}$.
As volume is a 3-dimensional we can express the coefficient of volume expansion in terms of coefficient of linear expansion as:
$\Rightarrow {\text{y = 3}}\alpha \\\ \Rightarrow {\text{y = 3 }} \times {\text{ 1}}{\text{.9 }} \times {\text{1}}{{\text{0}}^{ - 5}} \\\ \Rightarrow {\text{y = 5}}{\text{.7 }} \times {\text{ 1}}{{\text{0}}^{ - 5}}/^\circ {\text{C}}$
Coefficient of volume expansion of the silver rod is ${\text{5}}{\text{.7 }} \times {\text{ 1}}{{\text{0}}^{ - 5}}/^\circ {\text{C}}$.
Hence Option A is the right answer.

Note: In order to answer this type of questions the key is to know the concepts of linear expansion and volume expansion in solids. Identifying that volume expansion is a 3-dimensional form of linear expansion in solids is a crucial step in solving this problem.
Coefficients of linear and volume expansion denote the rate at which a material expands when subject to heat. The greater the value of coefficient of expansion, the more the material expands.