Halliday, Resnick & Walker — Temperature, Heat, and the First Law of Thermodynamics: Problem 99

Density change. The density \(\rho\) of something is the ratio of its mass \(m\) to its volume \(V\). If the volume is temperature dependent, so is the density. Show that a small change in density \(\Delta\rho\) with a small increase \(\Delta T\) in temperature is approximately given by \(\Delta\rho = -\beta\rho\Delta T\), where \(\beta\) is the coefficient of volume expansion. Explain the minus sign.