Question

A particular uranium alloy has a density of 18.75 g/cm3. Please answer the following questions below, providing the explanation to your answers. a. What volume is occupied by a critical mass of 49 kg of this alloy? b. The critical mass can be decreased to 16 kg if the alloy is surrounded by a layer of natural uranium (which acts as a neutron reflector). What is the volume of such smaller mass? Compare your answers to the approximate volumes of a baseball, a volleyball, and a basketball.

Answer 1

Answer:

• a) Critical mass of 49 kg of Uranium: 2600 cm³

• b) Critical mass of 16 kg Uranium: 850 cm³

Comparison:

• Baseball: 221 cm³

• Basketball: 7,238 cm³

• Baseball < Critical mass of 16 kg of uranium < critical mass of 49 kg of uranium < basketball.

Explanation:

To find the volumes of the two samples of uranium alloy, you must use the formula of density:

         $Density=\dfrac{mass}{volume}\\\\\\Volume=\dfrac{mass}{density}$

a) Critical mass of 49 kg

First, notice that 18.75g/cm³ is the same that 18.75 kg/dm³ (to convert from grams to kg you must divide by 1,000, and to convert from cm³ to dm³ you must divide by 1,000; thus, dividing both numerator an denominator by the same number leaves the fraction unchanged).

Now you can compute:

   $Volume=\dfrac{49kg}{18.75kg/dm^3}=2.6dm^3$

You can convert to cm³ multiplying by 1,000cm³/dm³.

Thus, volume = 2,600 cm³.

b) Critical mass = 16 kg

      $Volume=\dfrac{16kg}{18.75kg/dm^3}=0.85dm^3$

Volume = 850cm³

Compare with approximate volumes of a baseball, a volleyball, and a basketball:

You can calculate those volumes by using the formula for the volume of a sphere:

        $Volume=\dfrac{4}{3}\pi\times radius^3$

For the baseball: radius ≈ 3.75cm

      $Volume=\dfrac{4}{3}\pi\times (3.75cm)^3\approx 221cm^3$

For the basketball radius ≈ 12.0 cm

     $Volume=\dfrac{4}{3}\pi\times (12.0cm)^3approx 221cm^3\approx 7,238cm^3$

Thus, the comparison of the volumes is:

Baseball < Crittical mass 16 kg of Uranium 16 kg < Critical mass 49 kg of Uranium  < Basketball