(II) (a) Show that the change in the density ρof a substance, when the temperature changes by ΔT

step 1 Now, delta row, the change in density is final density minus initial density. That's mass by fin

(II) (a) Show that the change in the density ρof a...

(II) (a) Show that the change in the density $\rho$ of a substance, when the temperature changes by $\Delta T,$ is given by $\Delta \rho=-\beta \rho \Delta T .$ (b) What is the fractional change in density of a lead sphere whose temperature decreases from $25^{\circ} \mathrm{C}$ to $-55^{\circ} \mathrm{C} ?$

Key Concepts

Coefficient of Volume Expansion

This is a material property that quantifies how much the volume of a substance changes with a change in temperature. It is defined as the fractional increase in volume per unit change in temperature and is typically denoted by ? for volumetric expansion. Understanding ? is crucial for analyzing how temperature variations affect the dimensions and density of materials.

Density Dependence on Volume Change

Density is defined as the mass per unit volume of a substance. When the temperature changes, the volume of a material changes according to its coefficient of volume expansion, while its mass remains constant. Consequently, the change in density is directly related to the change in volume. This relationship is fundamental when deriving expressions that connect temperature change to changes in physical properties like density.

Fractional Change in Density

The fractional change in density refers to the relative change in a material’s density compared to its original density as a result of temperature-induced volume changes. By using the relationship between volume expansion and density, one can derive that the fractional change in density is approximately the negative of the coefficient of volume expansion times the temperature change. This concept allows for a straightforward estimation of how density will vary with temperature, which is especially useful in practical applications and calculations.

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