Question

The coefficient of volume expansion is:

Answer 1

In this question, we will use the basic relation between the linear expansion and change in temperature, also relation between the volume expansion and the change in temperature. Further, applying these formulas in the volume expression will give us the required result. Also, we will study the basics of Archimedes’s principle which tells us how the volume of liquid changes when a given mass is placed inside it.

Formula used:
$L \to L(1 + \alpha \Delta T)$
$V \to V(1 + \gamma \Delta T)$
$V = {L^3}$

Complete step by step solution:
As we know, the volume expansion is termed as the increase in the volume of the solid with increase in temperature i.e., on heating.
Let us take α and γ as the linear and volumetric expansion respectively, then linear expansion is given by:
$L \to L(1 + \alpha \Delta T)$
Similarly, volume expansion is given as:
$V \to V(1 + \gamma \Delta T)$
Here, we know that:
$V = {L^3}$
Putting the value of linear expansion in this equation, we will get:
$V = {L^3}{(1 + \alpha \Delta T)^3}$
$\Rightarrow V = {L^3}(1 + 3\alpha \Delta T + 2\alpha \Delta T + {\alpha ^2}\Delta {T^2} + {\alpha ^3}\Delta {T^3})$
Since the thermal expansion coefficient is of the order of parts per million per oC, we can neglect quadratic and cubic terms.
$V = {L^3}{(1 + \alpha \Delta T)^3}$
$\therefore \gamma = 3a$

Therefore, we get the required answer i.e., the coefficient of volume expansion is thrice of the coefficient of linear expansion.

Additional information:
As we know, volume is defined as the physical quantity of a three- dimensional space enclosed by a closed surface. If we take example: the space or shape occupied by the substance like solid, liquid, gas, or plasma.
Also, volume is often numerically given by using the S.I derived unit, called the cubic meter. This derived S.I unit for volume is the cubic meter based on the three dimensional use of the base unit for length, i.e., meter. The more common physical unit for volume is the liter.
Archimedes principle states that an object immersed in a fluid experiences some buoyant force that is equal in magnitude to the force of gravity on the displaced fluid. This law is also known as the law of buoyancy.
Now, as we know the weight of the displaced fluid will be equal to the subtraction of weight of object in vacuum and the weight of same object in fluid.
Also, the weight of the displaced fluid is directly proportional to the volume of the displaced fluid.
We should also know about the buoyant force. Buoyant force is an upward force which is exerted by a fluid on the object. Here, in the given fluid, the pressure increases with the given depth and this results in the weight of the overlying fluid. So, we can say that the pressure at the bottom of fluid is greater than the pressure at the top of the column of fluid.

Note:
Here one should notice that the Archimedes principle is only valid for fluids. So, we can say, where buoyant force can be observed. Also, the displaced fluid by the object is equal to the weight of the object immersed in the fluid. In the displaced fluid, gravity also plays an important role.