Maharashtra BoardEnglish MediumSTD 11 SciencePhysicsThermal Properties of Matter4 Marks
Question
Derive the relation between three coefficients of thermal expansion.
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Answer
Consider a square plate of side $l_0$ at $0 ^\circ\ C$ and h at $T ^\circ C$.
1. $\left.\right|_T=I_0(1+\alpha T)$
If area of plate at $0^{\circ} C$ is $A _0, A _0=l_0^2$
If area of plate at $T^{\circ} C$ is $A_T$,
$A _{ T }=l_{ T }^2=l_0^2(1+\alpha T )^2$
or $A_T=A_0(1+\alpha T)^2$
Also,
$ A_T=A_0(1+\beta T)^2$
${\left[\because \beta=\frac{ A _{ T }- A _0}{ A _0\left( T - T _0\right)}\right]}$
$ $ 2. Using Equations (1) and (2),
$ A _0(1+\alpha T )^2= A _0(1+\beta T )$
$\therefore 1+2 \alpha T +\alpha^2 T ^2=1+\beta T $
Since the values of a are very small, the term $\alpha ^2T^2$ is very small and may be neglected,$\therefore \beta = 2a$
The result is general because any solid can be regarded as a collection of small squares.
Relation between coefficient of linear expansion (α) and coefficient of cubical expansion (γ).
Consider a cube of side $l_0$ at $0 ^\circ C$ and $l_T$ at $T ^\circ C$.
$\therefore I_T=I_0(1+\alpha T)$
If volume of the cube at $0^{\circ} C$ is $V_0, V_0=l_0^3$
If volume of the cube at $T^{\circ} C$ is
$V _{ T }, V _{ T }=l_{ T }^3=l_0^3(1+\alpha T )^3$ $V _{ T }= V _0(1+\alpha)^3 \ldots \ldots \ldots .(1)$
Also,
$ V _{ T }= V _0(1+\gamma T) \ldots \ldots \ldots \ldots .(2)$
$\ldots \ldots \ldots \ldots . .\left[\because Y=\frac{ V _{ T }- V _0}{ V _0\left( T - T _0\right)}\right] $
2. Using Equations (1) and (2),
$ V_0(1+\alpha T)^3=V_0(1+\gamma T)$
$\therefore 1+3 \alpha T+3 \alpha^2 T^2+\alpha^3 T^3=1+\gamma T $
Since the values of a are very small, the terms with higher powers of a may be neglected,$\therefore γ = 3\alpha $
The result is general because any solid can be regarded as a collection of small cubes.
Relation between α, β and γ is given by,
$\alpha=\frac{\beta}{2}=\frac{\gamma}{3}$
where, $\alpha $ = coefficient of linear expansion.
$\beta$ = coefficient of superficial expansion,
$γ$ = coefficient of cubical expansion.
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