Question

A pycnometer is a device used to measure density. It weighs 20.455 g empty and 31.486 g when filled with water \(\left(d=1.00 \mathrm{~g} / \mathrm{cm}^{3}\right) .\) Pieces of an alloy are put into the empty, dry pycnometer. The mass of the alloy and pycnometer is \(28.695 \mathrm{~g} .\) Water is added to the alloy to exactly fill the pycnometer. The mass of the pycnometer, water, and alloy is \(38.689 \mathrm{~g}\). What is the density of the alloy?

Short answer

Answer: The density of the alloy is approximately \(7.946 \mathrm{~g/cm^3}\).

Step-by-step solution

Calculate the mass of water in the pycnometer

Subtract the mass of the empty pycnometer from the mass of the pycnometer filled with water: $$Mass_{water} = 31.486 \mathrm{~g} - 20.455 \mathrm{~g} = 11.031 \mathrm{~g}$$

Calculate the volume of water in the pycnometer

Divide the mass of water by its density to find the volume: $$Volume_{water} = \frac{Mass_{water}}{Density_{water}} = \frac{11.031 \mathrm{~g}}{1.00 \mathrm{~g/cm^3}} = 11.031 \mathrm{~cm^3}$$

Calculate the mass of the alloy

Subtract the mass of the empty pycnometer from the mass of the pycnometer containing the alloy: $$Mass_{alloy} = 28.695 \mathrm{~g} - 20.455 \mathrm{~g} = 8.240 \mathrm{~g}$$

Calculate the mass of water with the alloy in the pycnometer

Subtract the combined mass of the alloy and pycnometer from the mass of the pycnometer filled with water and alloy: $$Mass_{water\_with\_alloy} = 38.689 \mathrm{~g} - 28.695 \mathrm{~g} = 9.994 \mathrm{~g}$$

Calculate the volume of water displaced by the alloy

Divide the mass of displaced water by its density to find the volume: $$Volume_{displaced\_water} = \frac{Mass_{water\_with\_alloy} - Mass_{water}}{Density_{water}} = \frac{9.994 \mathrm{~g} - 11.031 \mathrm{~g}}{1.00 \mathrm{~g/cm^3}} = -1.037 \mathrm{~cm^3}$$ Since the volume of displaced water is the same as the volume of the alloy, the volume of the alloy is: $$Volume_{alloy} = -Volume_{displaced\_water} = 1.037 \mathrm{~cm^3}$$

Calculate the density of the alloy

Divide the mass of the alloy by its volume to find its density: $$Density_{alloy} = \frac{Mass_{alloy}}{Volume_{alloy}} = \frac{8.240 \mathrm{~g}}{1.037 \mathrm{~cm^3}} = 7.946 \mathrm{~g/cm^3}$$ The density of the alloy is approximately \(7.946 \mathrm{~g/cm^3}\).