Extra Credit 10 points A device commonly used to power the e

Extra Credit (10 points): A device commonly used to power the electronic components on rovers that are sent to explore Mars is the radioisotope thermoelectric generator (RTG). The isotope most commonly used for this purpose is 238Pu due to its half-life of 87.7 years and its power output of approximately 4.4 kW. Typically, 7.8 kg of 238Pu is used for a single RTG. A group of students working with a national laboratory is attempting to produce enough 238Pu to make an RTG for their senior design project. They decided that the most effective method to do this is through the radioactive decay of 242Cm which has a half-life of 162.8 days and is purchased from a vendor How many kilograms of 242Cm do they need to make enough 238Pu in 45 days to build the RTG? Hint: This problem involves decay and production

Solution

²⁴²Cm----------->²³⁸Pu + ⁴He

Amount of Pu required is 7.8 kg = 7800 g

Amount of Cm required to produces 7.8 kg of Pu after 45 days

N = Noe^-t

238 g of Pu has 6.023x10²³ nuclei

Number of Pu nuclei required is 7800 x 6.023x10²³ / 238 = 1.97x10²⁵ (from ²⁴²Cm)

N = 1.97x10²⁵

N_o = Amount of Cm required

= 0.693 / 162.8 day^-1

t = 45 days

N = N_oe^-t

1.97x10²⁵ = N_oe^{(-0.693 x45 / 162.8)}

1.97x10²⁵ = N_oe^(-0.191)

1.97x10²⁵ = N_o x 0.826

N_o = 1.97x10²⁵ / 0.826 = 2.38x10²⁵

242 g of Cm contains 6.023x10²³ nuclei

so 2.38x10²⁵ nuclei corresponds to 9562 g = 9.562 kg

So the amount of ²⁴²Cm required to make enough ²³⁸Pu after 45 days is 9.562 kg