Question

The density of a lead at $0^\circ {\text{C}}$ is ${\text{11}}{\text{.34}}\,{\text{gm/cc}}$. What is the density of lead at $100^\circ {\text{C}}$, given the coefficient of linear expansion of lead =$28 \times {10^{ - 6}}/^\circ {\text{C}}$:
A. ${\text{13}}{\text{.25}}\,{\text{g/c}}{{\text{m}}^3}$
B. ${\text{17}}{\text{.25}}\,{\text{g/c}}{{\text{m}}^3}$
C. ${\text{18}}{\text{.25}}\,{\text{g/c}}{{\text{m}}^3}$
D. ${\text{11}}{\text{.25}}\,{\text{g/c}}{{\text{m}}^3}$

Answer 1

Use the formula for thermal expansion of a material in terms of its density. This formula gives the relation between initial density of material, changed density of material, coefficient of volumetric thermal expansion and change in temperature of material. Also, use the formula for volumetric thermal expansion coefficient in terms of linear thermal expansion coefficient.

Formula used:
The expression for thermal expansion of a material in terms of its density is given by
$\rho = {\rho _0}\left( {1 - \gamma \Delta T} \right)$ …… (1)
Here, ${\rho _0}$ is the initial or original density of the material, $\rho$ is the changed density of material, $\gamma$ is the coefficient of volumetric thermal expansion and $\Delta T$ is the change in temperature of the material.
The volumetric thermal expansion coefficient $\gamma$ is
$\gamma = 3\alpha$
Here, $\alpha$ is the coefficient of linear thermal expansion.

Complete step by step answer:
We have given that the initial density of lead at temperature $0^\circ {\text{C}}$ is ${\text{11}}{\text{.34}}\,{\text{gm/cc}}$.
${T_i} = 0^\circ {\text{C}}$
${\rho _0} = {\text{11}}{\text{.34}}\,{\text{gm/cc}}$
We have asked to determine the changed density $\rho$ of lead at the temperature $100^\circ {\text{C}}$.
${T_f} = 100^\circ {\text{C}}$
The coefficient of linear thermal expansion for lead is $28 \times {10^{ - 6}}/^\circ {\text{C}}$.
$\alpha = 28 \times {10^{ - 6}}/^\circ {\text{C}}$
We should first determine the temperature difference between two temperatures of lead.
$\Delta T = {T_f} - {T_i}$

Substitute $100^\circ {\text{C}}$ for ${T_f}$ and $0^\circ {\text{C}}$ for ${T_i}$ in the above equation.
$\Delta T = \left( {100^\circ {\text{C}}} \right) - \left( {0^\circ {\text{C}}} \right)$
$\Rightarrow \Delta T = 100^\circ {\text{C}}$
Hence, the change in temperature of the lead is $100^\circ {\text{C}}$.
Let us now determine the volumetric coefficient of thermal expansion.
Substitute $28 \times {10^{ - 6}}/^\circ {\text{C}}$ for $\alpha$ in equation (2).
$\gamma = 3\left( {28 \times {{10}^{ - 6}}/^\circ {\text{C}}} \right)$
$\Rightarrow \gamma = 84 \times {10^{ - 6}}/^\circ {\text{C}}$
Hence, the coefficient of volumetric thermal expansion is $84 \times {10^{ - 6}}/^\circ {\text{C}}$.
Let us determine the changed density of lead using equation (1).
Substitute ${\text{11}}{\text{.34}}\,{\text{gm/cc}}$ for ${\rho _0}$, $84 \times {10^{ - 6}}/^\circ {\text{C}}$ for $\gamma$ and $100^\circ {\text{C}}$ for $\Delta T$ in equation (1).
$\rho = \left( {{\text{11}}{\text{.34}}\,{\text{gm/cc}}} \right)\left[ {1 - \left( {84 \times {{10}^{ - 6}}/^\circ {\text{C}}} \right)\left( {100^\circ {\text{C}}} \right)} \right]$
$\Rightarrow \rho = \left( {{\text{11}}{\text{.34}}\,{\text{gm/cc}}} \right)\left[ {1 - 84 \times {{10}^{ - 4}}} \right]$
$\Rightarrow \rho = \left( {{\text{11}}{\text{.34}}\,{\text{gm/cc}}} \right)\left[ {1 - 0.0084} \right]$
$\Rightarrow \rho = \left( {{\text{11}}{\text{.34}}\,{\text{gm/cc}}} \right)\left( {0.9916} \right)$
$\Rightarrow \rho = {\text{11}}{\text{.24}}\,{\text{g/c}}{{\text{m}}^3}$
$\Rightarrow \rho \approx {\text{11}}{\text{.25}}\,{\text{g/c}}{{\text{m}}^3}$

Therefore, the density of lead is ${\text{11}}{\text{.25}}\,{\text{g/c}}{{\text{m}}^3}$.

So, the correct answer is “Option D”.

Note:
The students may forget to multiply the coefficient of linear thermal expansion by 3. As the density of the lead has changed, the volume of the lead also changes. Hence, the term coefficient of linear thermal expansion is replaced by volumetric thermal expansion coefficient which is thrice the linear thermal expansion coefficient.