Question

When a liquid is heated in a copper vessel its coefficient of apparent expansion is$6 \times {10^{ - 6}}{/^0}C$ . When the same liquid is heated is heated in a steel vessel its coefficient of apparent expansion is \[24 \times {10^{ - 6}}{/^0}C\].If the coefficient of linear expansion for copper is $18 \times {10^{ - 6}}{/^0}C$ , the coefficient of linear expansion for steel is:
A. \[20 \times {10^{ - 6}}{/^0}C\]
B. \[24 \times {10^{ - 6}}{/^0}C\]
C. \[34 \times {10^{ - 6}}{/^0}C\]
D. \[12 \times {10^{ - 6}}{/^0}C\]

Answer 1

Hint - To give the answer of this question, we use the theory of thermal expansion. First we see the definition of thermal expansion and also we define the coefficient of linear thermal expansion. Discuss about coefficient and relation between them. Here we use formula ${\gamma _{real}}$ =${\gamma _{app}}$+${\gamma _{container}}$; where ${\gamma _{real}}$ is coefficient of volumetric expansion of vessel and liquid, ${\gamma _{app}}$is coefficient of volumetric expansion of liquid, ${\gamma _{container}}$ is coefficient of volumetric expansion of vessel.

Complete answer:
Thermal expansion is the tendency of matter to change its shape, density, area and volume when changing its temperature.
The relative expansion divided by the change in temperature is called the coefficient of linear thermal expansion. It varies with temperature.
 There are three types of coefficient; volumetric, aerial and linear coefficient.
\[\alpha \]is the coefficient of linear expansion, \[\beta \]is the coefficient of areal expansion and $\gamma $ is the coefficient of volumetric expansion.
Relation between them is;
\[\alpha \]:\[\beta \]:$\gamma $=1:2:3
$\therefore $ $\gamma $=3\[\alpha \]
In this question, it is given that;
${\gamma _{app}}$(Cu) =$6 \times {10^{ - 6}}{/^0}C$, ${\alpha _{container}}$(Cu) =$18 \times {10^{ - 6}}{/^0}C$, ${\gamma _{app}}$(S) =\[24 \times {10^{ - 6}}{/^0}C\] ,${\alpha _{container}}$(S) =?
${\gamma _{container}}$(Cu) =3${\alpha _{container}}$(Cu) = $3 \times 18 \times {10^{ - 6}}{/^0}C$
                                     =$54 \times {10^{ - 6}}{/^0}C$
 Putting the value in above equation, we get;
                        ${\gamma _{real}}$ =${\gamma _{app}}$+${\gamma _{container}}$
So, ${\gamma _{app}}$(Cu) + ${\gamma _{container}}$(Cu) = ${\gamma _{app}}$(S) +${\gamma _{container}}$(S)
                     $6 \times {10^{ - 6}}{/^0}C$+$54 \times {10^{ - 6}}{/^0}C$=\[24 \times {10^{ - 6}}{/^0}C\]+3${\alpha _{container}}$(S)
                            $60 \times {10^{ - 6}}{/^0}C$ =\[24 \times {10^{ - 6}}{/^0}C\]+3${\alpha _{container}}$(S)
                                                3${\alpha _{container}}$(S) =$36 \times {10^{ - 6}}{/^0}C$
                                      ${\alpha _{container}}$(S) = $12 \times {10^{ - 6}}{/^0}C$
Hence, the coefficient of linear expansion for steel is$12 \times {10^{ - 6}}{/^0}C$. Option D is correct.

Note –The volumetric thermal expansion coefficient is the most thermal expansion coefficient, and the most relevant for liquids. \[\alpha \]:\[\beta \]:$\gamma $=1:2:3 this relation occurs for isotropic material. Some examples of thermal expansion are engine coolant; we use it in car radiators, expansion joints it’s used in bridges; bridges have a long span and in hot weather the materials that the bridge is made of will have an expansion effect.