Question

A crystal has linear coefficient of expansion 9*10^-5, 12*10^-5, 7*10^-5/K along 3 mutually perpendicular directions the volume expansion coefficient is

Answer 1

Answer:

Let the sides be L1, L2, L3   with volume V =L1 * L2 * L3

and the corresponding linear coefficients of expansion a, b, c

The area of sides L1 * L2 after expansion is

A2 = (L1 + a L1) * (L2 * b L2) = L1 L2 + a L1 L2 + b L1 L2 + a b L1 L2

or A2 = L1 L2 + (a + b) L1 L2)   since a b L1 L2 is negligble

V2 = A2 ( L3 + c L3)

Multiplying and simplifying  gives

V2 = L1 L2 L3 + L1 L2 L3 (a + b + c)  = V + V (a + b + c)

where terms such as (a c + b c) L1 L2 L3  have been dropped

This gives V2 - V = V (a + b +c)

and the coefficient of volume expansion is

(9 + 12 + 7) * 10E-5 /K = 28 * 10E-5 /K