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Question: When a liquid is heated in a copper vessel its coefficient of apparent expansion is\(6 \times {10^{ ...

When a liquid is heated in a copper vessel its coefficient of apparent expansion is6×10−6/0C6 \times {10^{ - 6}}{/^0}C . When the same liquid is heated is heated in a steel vessel its coefficient of apparent expansion is 24×10−6/0C24 \times {10^{ - 6}}{/^0}C.If the coefficient of linear expansion for copper is 18×10−6/0C18 \times {10^{ - 6}}{/^0}C , the coefficient of linear expansion for steel is:
A. 20×10−6/0C20 \times {10^{ - 6}}{/^0}C
B. 24×10−6/0C24 \times {10^{ - 6}}{/^0}C
C. 34×10−6/0C34 \times {10^{ - 6}}{/^0}C
D. 12×10−6/0C12 \times {10^{ - 6}}{/^0}C

Explanation

Solution

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 γreal{\gamma _{real}} =γapp{\gamma _{app}}+γcontainer{\gamma _{container}}; where γreal{\gamma _{real}} is coefficient of volumetric expansion of vessel and liquid, γapp{\gamma _{app}}is coefficient of volumetric expansion of liquid, γcontainer{\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.
α\alpha is the coefficient of linear expansion, β\beta is the coefficient of areal expansion and γ\gamma is the coefficient of volumetric expansion.
Relation between them is;
α\alpha :β\beta :γ\gamma =1:2:3
∴\therefore γ\gamma =3α\alpha
In this question, it is given that;
γapp{\gamma _{app}}(Cu) =6×10−6/0C6 \times {10^{ - 6}}{/^0}C, αcontainer{\alpha _{container}}(Cu) =18×10−6/0C18 \times {10^{ - 6}}{/^0}C, γapp{\gamma _{app}}(S) =24×10−6/0C24 \times {10^{ - 6}}{/^0}C ,αcontainer{\alpha _{container}}(S) =?
γcontainer{\gamma _{container}}(Cu) =3αcontainer{\alpha _{container}}(Cu) = 3×18×10−6/0C3 \times 18 \times {10^{ - 6}}{/^0}C
=54×10−6/0C54 \times {10^{ - 6}}{/^0}C
Putting the value in above equation, we get;
γreal{\gamma _{real}} =γapp{\gamma _{app}}+γcontainer{\gamma _{container}}
So, γapp{\gamma _{app}}(Cu) + γcontainer{\gamma _{container}}(Cu) = γapp{\gamma _{app}}(S) +γcontainer{\gamma _{container}}(S)
6×10−6/0C6 \times {10^{ - 6}}{/^0}C+54×10−6/0C54 \times {10^{ - 6}}{/^0}C=24×10−6/0C24 \times {10^{ - 6}}{/^0}C+3αcontainer{\alpha _{container}}(S)
60×10−6/0C60 \times {10^{ - 6}}{/^0}C =24×10−6/0C24 \times {10^{ - 6}}{/^0}C+3αcontainer{\alpha _{container}}(S)
3αcontainer{\alpha _{container}}(S) =36×10−6/0C36 \times {10^{ - 6}}{/^0}C
αcontainer{\alpha _{container}}(S) = 12×10−6/0C12 \times {10^{ - 6}}{/^0}C
Hence, the coefficient of linear expansion for steel is12×10−6/0C12 \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. α\alpha :β\beta :γ\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.