Question

Some textbooks use the unit \(\mathrm{K}^{-1}\) rather than \({ }^{\circ} \mathrm{C}^{-1}\) for values of the linear expansion coefficient; see Table \(17.2 .\) How will the numerical values of the coefficient differ if expressed in \(\mathrm{K}^{-1}\) ?

Short answer

Answer: The numerical values of the linear expansion coefficient do not differ when expressed in Kelvin inverse (K^{-1}) and Celsius inverse (°C^{-1}). This is because the incremental changes in both temperature scales are the same.

Step-by-step solution

Understand the relation between Kelvin and Celsius scales

The difference between the Kelvin and Celsius scales is the zero point. In the Celsius scale, 0 °C is the freezing point of water and 100 °C is the boiling point. On the Kelvin scale, 0 K is absolute zero and 273.15 K is the freezing point of water. However, the incremental change in one unit remains the same between these two scales (1 °C change equals 1 K change). In other words, the change in Kelvin and the change in Celsius are equal.

Compare the numerical values of the linear expansion coefficient

Now that we know a change in Celsius is equal to a change in Kelvin, let's consider the numerical values of the linear expansion coefficient in both units. The linear expansion coefficient, α, has the units of inverse length times temperature (L^{-1}T^{-1}). When expressed in Celsius inverse, we can write it as α(°C^{-1}), and when expressed in Kelvin inverse, it's written as α(K^{-1}). Since 1 °C change equals 1 K change, it means that the values of the linear expansion coefficient when expressed in Kelvin inverse (K^{-1}) will be the same as when expressed in Celsius inverse (°C^{-1}). That is: α(K^{-1}) = α(°C^{-1})

Conclusion

In conclusion, the numerical values of the linear expansion coefficient do not differ when expressed in Kelvin inverse (K^{-1}) and Celsius inverse (°C^{-1}). The relation between the two units reflects that the incremental changes in both temperature scales are the same.