Question

# If α,β and γ are the coefficient of linear, areal and volume expansion respectively, then which of the following options correctly represent the relation between them?γ=3αα=3γβ=3αγ=3β

Solution

## The correct option is A γ=3αLet us consider a three dimensional solid having a volume expansion coefficient (γ). Let the original volume of the solid be (V). Here we assume, Volume V=L3. Then by heating the solid by changing the temperature (ΔT), the new volume of the solid can be written as, V+ΔV=(L+ΔL)3    ............(1) where (L+ΔL) is the new length of the solid along each direction in 3-dimensional space. Using binomial expansion on (1), we get V+ΔV=(L3+(ΔL)3+3L2ΔL+3L(ΔL)2) From the definition of linear and volume expansions ΔL=Lα ΔT and ΔV=Vγ ΔT ⇒V+ΔV=L3+(LαΔT)3+3L2(LαΔT)+3L(LαΔT)2 ⇒V+Vγ ΔT=L3+L3(α3ΔT3+3α2ΔT2+3α ΔT) [Neglecting α2 and α3 terms as α is very small and using V=L3] ⇒VγΔT=L3×3αΔT we get, γ =3α Hence, option (a) is the correct answer.

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