Question

Which of the following equations best states the relationship between a coefficient of volume expansion $\beta$ and its coefficient of linear expansion $\alpha$?

• A. $\beta =\alpha$
• B. $\beta =3\alpha$
• C. $\beta =\alpha +{\alpha }^2$
• D. $\beta ={\alpha }^3$
• E. $\beta =3{\alpha }^3$

Answer 1

The correct option is B $\beta =3\alpha$

For linear expansion, $L_T=L_0[1+\alpha \Delta T]...\left(1\right)$

For volume expansion, $V_T=V_0[1+\beta \Delta T]......\left(2\right)$

Now, $V_T=L_T^3$ and $V_0=L_0^3$

So, $V_0[1+\beta \Delta T]=\left[L_0\right(1+\alpha \Delta T){]}^3\sim L_0^3[1+3\alpha \Delta T]$ (by neglecting the higher order term )

or $V_0[1+\beta \Delta T]=V_0[1+3\alpha \Delta T]$

Thus, $\beta =3\alpha$