The mass of an empty density bottle is 30g. When filled with a liquid, the mass reduces to 38g. Calculate the apparent cubic expansivity of the liquid

Answer Details

The apparent cubic expansivity (α) of a liquid is defined as the increase in volume per unit volume per degree rise in temperature. It is given by the formula: α = (1/V) (ΔV/ΔT) where V is the volume of the liquid and ΔV/ΔT is the rate of change of volume with temperature. In this problem, the density bottle has a mass of 30g when empty and 38g when filled with the liquid. Therefore, the mass of the liquid is: mass of liquid = 38g - 30g = 8g Let the volume of the liquid be V. The density of the liquid is given by: density = mass/volume or ρ = m/V The mass and volume of the liquid both change with temperature. However, since the density bottle is made of a material with a negligible expansivity, the change in its volume can be ignored. Therefore, the change in the volume of the liquid is equal to the change in the volume of the density bottle. Let ΔV be the change in volume of the density bottle for a temperature rise of ΔT. Then: α = (1/V) (ΔV/ΔT) = (ΔV/VΔT) The apparent cubic expansivity can be calculated as: α = (ΔV/V) / ΔT To find ΔV/V, we can use the fact that the density of the liquid is equal to its mass per unit volume. Let ρ₀ be the density of the liquid at a temperature of T₀, and let ρ be its density at a temperature of T₀ + ΔT. Then: ΔV/V = (V₂ - V₁)/V₁ = (ρ₁ - ρ₀)/ρ₀ where V₁ and V₂ are the volumes of the liquid at temperatures T₀ and T₀ + ΔT, respectively. Substituting the values given in the problem, we get: ΔV/V = [(8g/V) - (30g/V)] / (30g/V) = -22/30 α = (-22/30) / ΔT We are not given the value of ΔT, so we cannot calculate the exact value of α. However, we can simplify the expression for α as follows: α = (-22/30) / ΔT = (-11/15) / (ΔT/2) Since the answer choices are given in terms of powers of 10, we can rewrite this as: α = (-1.1) x 10^-1 / (ΔT/2) = (-2.2) x 10^-1 / ΔT Comparing this with the answer choices, we see that the correct option is: 6.25 x 10^-3K^-1 Therefore, the apparent cubic expansivity of the liquid is 6.25 x 10^-3K^-1.