Question

A uniform pressure $p$ exerted on all sides of a solid cube at temperature $0^\circ \,{\text{C}}$. In order to bring the volume of the cube to the original volume, the temperature of the cube must be increased by $t^\circ \,{\text{C}}$. If $\alpha$ is the linear coefficient of thermal expansion and $K$ is the bulk modulus of the material of the cube, then $t$ is equal to
A. $\dfrac{{3P}}{{K\alpha }}$
B. $\dfrac{P}{{2\alpha K}}$
C. $\dfrac{P}{{3\alpha K}}$
D. $\dfrac{P}{{\alpha K}}$

Answer 1

Use the formulae for the bulk strain, linear expansion of the material and the relation between the linear strain and bulk strain. These equations give the relation between the change in volume, change in length of the edge, original length, original volume, bulk modulus, thermal expansion coefficient and pressure on the cube.

Formula used:
The bulk modulus $K$ of a material is given by
$K = \dfrac{P}{{\dfrac{{\Delta V}}{V}}}$ …… (1)
Here, $P$ is the bulk stress of the material, $\Delta V$ is the change in volume of the material and $V$ is the original volume of the material.
The change in the length $\Delta l$ of a material due to thermal expansion is
$\Delta l = {l_0}\alpha \Delta T$ …… (2)
Here, is the original length of the material, is the linear thermal coefficient and is the change in the temperature.
The relation between the bulk strain and linear strain is
${\text{Bulk strain}} = 3\left( {{\text{Linear strain}}} \right)$ …… (3)

Complete step by step answer:
A uniform pressure $p$ exerted on all sides of a solid cube at temperature $0^\circ \,{\text{C}}$. In order to bring the volume of the cube to the original volume, the temperature of the cube is increased by $t^\circ \,{\text{C}}$.
Rewrite equation (1) for the linear thermal expansion of the cube.
$\Delta l = {l_0}\alpha t$
Here, ${l_0}$ is the original length of the edge of the cube, $\Delta l$ is the change in length of edge of the cube and $t$ is the change in temperature of the cube.
Rearrange the above equation for $\dfrac{{\Delta l}}{{{l_0}}}$.
$\dfrac{{\Delta l}}{{{l_0}}} = \alpha t$
Rewrite equation (1) for the bulk modulus $K$ of the material of the cube.
$K = \dfrac{P}{{\dfrac{{\Delta V}}{{{V_0}}}}}$
Here, ${V_0}$ is the original volume of the cube.
Rearrange the above equation for $\dfrac{{\Delta V}}{{{V_0}}}$.
$\dfrac{{\Delta V}}{{{V_0}}} = \dfrac{P}{K}$
The linear strain of the cube is the ratio of the change in length $\Delta l$ of the edge of the cube to the original length ${l_0}$ of the edge of the cube.
${\text{Linear strain}} = \dfrac{{\Delta l}}{{{l_0}}}$
The bulk strain of the cube is the ratio of the change in volume $\Delta V$ of the cube to the original volume ${V_0}$ of the cube.
${\text{Bulk strain}} = \dfrac{{\Delta V}}{{{V_0}}}$
Substitute $\dfrac{{\Delta l}}{{{l_0}}}$ for ${\text{Linear strain}}$ and $\dfrac{{\Delta V}}{{{V_0}}}$ for ${\text{Bulk strain}}$ in equation (3).
$\dfrac{{\Delta V}}{{{V_0}}} = 3\dfrac{{\Delta l}}{{{l_0}}}$
Substitute $\alpha t$ for $\dfrac{{\Delta l}}{{{l_0}}}$ in the above equation.
$\dfrac{{\Delta V}}{{{V_0}}} = 3\alpha t$
Substitute $\dfrac{P}{K}$ for $\dfrac{{\Delta V}}{{{V_0}}}$ in the above equation.
$\dfrac{P}{K} = 3\alpha t$
Rearrange the above equation for the change in temperature $t$.
$t = \dfrac{P}{{3\alpha K}}$
Therefore, the change in temperature must be $\dfrac{P}{{3\alpha K}}$.
Hence, the correct option is C.

So, the correct answer is “Option C”.

Note:
Since the volume of a cube is three times the length of the edge of the cube.
The volume strain for the cube is three times the linear strain of the cube.