Question

A metal is heated from $0^{\circ }C$ to ${500}^{\circ }C$ and its density reduces to $1/1.027$ of its original density. Determine the coefficient of linear expansion for this metal, considering it constant for the given range of temperature.

Answer 1

We know that volume coefficient is thrice of linear coefficient '

$\gamma =3\alpha$

Let the density at 0 be $\rho =\frac{m}{V}$

Now when heated to 500 density becomes

${\rho }_1=\frac{\rho }{1.027}=\frac{m}{1.027V}$ ......1

Change in volume $\Delta V=V\times \gamma \Delta T$

$\Delta V=V\times 3\alpha \times 500$

New volume $V_1=V+\Delta V$

${\rho }_1=\frac{m}{V_1}=\frac{m}{V+V\times 3\alpha \times 500}$ ...(2)

Equating equation 1 and 2, $\frac{m}{1.027V}=\frac{m}{V+V\times 3\alpha \times 500}$

$\alpha =1.8\times {10}^{-5}{}^o{C}^{-1}$