Question

Question 10: The units for the coefficient of linear expansion \({\bf{\alpha }}\) are \({\left( {{\bf{C^\circ }}} \right)^{{\bf{ - 1}}}}\)and there is no mention of a length unit such as meters. Would the expansion coefficient change if we used feet or millimeters instead of meters? Explain.

Short answer

The coefficient of expansion would not change if we use feet or millimeters instead of meters as the units of length would cancel out in the expression of the coefficient of linear expansion.

Step-by-step solution

Linear expansion

Most of the solid substances linearly expand on heating. When the temperature of a solid object of length\({l_0}\)is increased by \(\Delta T\), the length of the object increases by \(\Delta l\) amount. The change in the length of the object is given as:

\(\Delta l = \alpha {l_0}\Delta T\) ... (i)

Here, \(\alpha \) is known as the coefficient of linear expansion.

Discussion on units of coefficient of linear expansion

Using equation (i), the expression for the coefficient of linear expansion can be written as:

\(\alpha = \frac{{\Delta l}}{{{l_0}\Delta T}}\)

It is clear from the above expression that the units of length cancel out, i.e., the unit of \(\Delta l\) cancels out with the unit of \({l_0}\).Therefore, there is no mention of length units, and the units of coefficient of linear expansion are \({\left( {{\rm{C}}^\circ } \right)^{ - 1}}\).

Thus, the coefficient of linear expansion would not change on changing the units of length if the same units of length are used both for \(\Delta l\)and \({l_0}\).So, we can use feet or millimeters instead of meters as the units of length in the expression for coefficient of linear expansion.