Answer to Question #245762 in Molecular Physics | Thermodynamics for Cat

Question

For an isotropic solid, show that the coefficients of expansion α, 𝛽 and 𝛾 are related as

(a) 𝛽 = 2𝛼

(b) 𝛾 = 3?

Answer 1

We know that linear expansion

$l_1 = l(1+\alpha \Delta T)...(i)$

Area expansion

$A_1=A(1+\beta \Delta T)=l^2(1+\beta \Delta T)...(ii)$

Volumetric expansion

$V_1= V(1+\gamma \Delta T)...(iii)$

But we know that area $(A_1)=l_{length}\times l_{base}=l^2(1+\alpha \Delta T)^2$

Now, taking the bi-nominal expansion,

$A_1=l^2(1+2\alpha)...(iv)$

from equation (ii) and (iv)

So, $\beta = 2\alpha$

Similarly,

$V_1= A_1\times l_1=l^2(1+3\alpha)...(v)$

Hence, from equation (i) and (v)

So, $\gamma = 3\alpha$