Radioactive decay and Radium extraction calculation

[Civy_MEO]

Homework Statement

A Uranium mine extracts sandstone in which the Uranium ore grade is 0.25%. Consideration is given to extracting Radium from the tailings. Estimate the concentration of Radium in the tailings and, using the data provided, determine if the proposal is financially viable.

Homework Equations

Assume Uranium and it's daughter product are in secular equlibrium.
Half-life of U238 = 4.47 x 10^9 years
Half-life of Ra266 = 1600 years
Cost of Radium Extraction = $10 per tonne of tailings
Radium Market price = $5000 per gram

The Attempt at a Solution

My attempt would be to divide the half lives to show that for every atom of Ra there is 2.79 Million atoms of U.

e.g. 4.47 x 10^9 / 1600 = 2.79 x 10^6

1/2.79 x 10^6 = 3.70 x 10^-7, number of atoms of Ra for each atom of U

Then calculate that for 1tonne of sandstone 0.25% will be Uranium

1tonne x 0.0025 = 2.5kg of U

Of which 3.70 x 10^-7 x 2500 = 9.25 x 10^-7 kilograms will be Radium.
(which is 9.25 x 10^-4 grams)

1 / 9.25 x 10^-4 = 1082 number of tonnes required to accumulate 1 g of Radium

Cost = 1082 x $10 = $10820 per gram, compared to market value of $5k this is not finanacially viable.

I would appreciate it if someone could check my workings, one of my colleagues mentioned that he used Avagadro's number to calculate a soloution. Do you believe this to be a correct soloution?!

Many thanks

 

[LowlyPion]

Welcome to PF.

Your secular equilibrium relationship looks to directly yield the ratio of the weights of each that can be recovered from the sample, so I'm not sure that worrying too much about the relative masses or figuring molar numbers with the weights being so similar 238 and 226, really affects the return on investment calculation all that much.

If it is to be economical, you should turn a profit on buying 500 tonnes if you can extract 1 gram of radium. But as you found you only can extract .0009g per tonne of tailings, or about half that.

 

[nlightner]

for your help!

Your approach to the problem is correct. Using the information provided, it can be calculated that for every 1 tonne of sandstone, there will be 9.25 x 10^-4 grams of Radium. This means that to accumulate 1 gram of Radium, 1082 tonnes of sandstone would need to be processed.

As for your colleague's suggestion of using Avogadro's number, that would not be necessary in this calculation as we are dealing with macroscopic amounts of material (tonnes) rather than individual atoms. Avogadro's number is typically used for calculating the number of atoms or molecules in a given mass of a substance.

In terms of financial viability, it is clear that the cost of extracting Radium from the tailings would far exceed its market value. Therefore, the proposal is not financially viable. Additional factors such as the cost of processing and the potential environmental impacts would also need to be considered in making a final decision.