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We wish to determine the density of a metallic alloy at 25oC. This is done as follows. A pycnometer weighs 20.455g empty and 31.486g when filled with water at 25oC. The density of water at 25oC is 0.99704 g/mL. Pieces of the alloy are put into the empty, dry pycnometer. The mass of the alloy and the pycnometer is 28.695g. Water is added to the pycnometer to exactly fill it. The mass of the entire system is now 38.689g. What is the density of the alloy

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From the pycnometer that weighs 20.455g empty and 31.486g when filled with water at 25oC; we can infer that the density of the alloy in this system is 7.89 g/mL

From the parameters given;

  • the mass of an empty pycnometer = 20.455 g
  • when the pycnometer is filled with water, the mass = 31.486 g

To determine the mass of the water, we have:

  • Mass of water = mass of pycnometer with water - mass of empty pycnometer
  • Mass of water = (31.486 - 20.455) g
  • Mass of water = 11.031 g

However, we need to find the volume of the water, since density is given as 0.99704 g/mL

Now, using the relation:

[tex]\mathbf{Density = \dfrac{mass}{volume}}[/tex]

[tex]\mathbf{0.99704 g/mL = \dfrac{11.031 g}{volume}}[/tex]

[tex]\mathbf{volume = \dfrac{11.031 \ g}{0.99704 \ g/mL}}[/tex]

[tex]\mathbf{volume \ of \ water = \dfrac{11.031 \ g}{0.99704 \ g/mL}}[/tex]

volume of water = 11.064 mL

Since the volume of the water is equivalent to the volume of the pycnometer;

  • we can infer that the volume of the pycnometer = 11.064 mL

Similarly, given that:

  • mass of pycnometer and alloy = 28.695 g
  • mass of the entire system (i.e. pycnometer, alloy & water) = 38.689 g

∴

  • Mass of the water = 38.689 g - 28.695 g
  • Mass of the water = 9.994 g

Here, from the entire system, the volume of the water again is calculated as follows:

[tex]\mathbf{Density = \dfrac{mass}{volume}}[/tex]

[tex]\mathbf{0.99704 g/mL = \dfrac{9.994 g}{volume}}[/tex]

[tex]\mathbf{volume = \dfrac{9.994 \ g}{0.99704 \ g/mL}}[/tex]

[tex]\mathbf{volume \ of \ water = \dfrac{11.031 \ g}{0.99704 \ g/mL}}[/tex]

Volume of water = 10.02 mL

Finally, by applying the volume of the pycnometer as well as the volume of the water to determine the volume of the alloy, we have:

  • Volume of alloy = volume of pycnometer - volume of water
  • Volume of alloy =  11.064 mL - 10.02 mL
  • Volume of alloy = 1.044 mL

Recall that:

  • the mass of the empty pycnometer = 20.455 g, and;
  • the mass of the pycnometer and alloy = 28.695 g

From the parameters above, we can deduce the mass of the alloy as follows;

Mass of alloy = 28.695 g - 20.455 g

Mass of alloy = 8.24 g

We know now that:

[tex]\mathbf{Density = \dfrac{mass}{volume}}[/tex]

∴

[tex]\mathbf{Density \ of \ the \ alloy = \dfrac{8.24 \ g}{1.044 \ mL}}[/tex]

Density of the alloy = 7.89 g/mL

Therefore, from the above calculation, we can conclude that the density of the alloy = 7.89 g/mL

Learn more about density, mass and volume here:

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