Problem 2

Astatine is the rarest naturally occurring element in the Earth's crust, with a total estimated mass of 30 g at any point in time; assume all astatine is the most stable naturally occurring isotope

^{219}

At. (a) Estimate how many At atoms occur naturally on Earth at any point in time. (b) Bismuth, obtained as a by-product of lead purification, is concentrated by adding magnesium and calcium to a lead–bismuth mixture; bismuth forms alloy A containing 7.91 % Ca and 9.60 % Mg by mass, with no lead. Determine the empirical formula of alloy A. (c) Alloy A reacts with chlorine gas to form bismuth and two ionic compounds B and C in a 2:1 ratio. Write the formulae of B and C. (d) Astatine is made by bombarding

^{209}_{83}

Bi with

\alpha

particles; the nuclei fuse to form

^{x}_{y}

At and two neutrons. Write the mass and atomic numbers of the astatine isotope. (e) Excess bismuth is removed with concentrated nitric acid, forming bismuth(III) nitrate, water and a colourless gas of molar mass 30.01 g mol

^{-1}

. Write the equation. (f) BrAt is made in two steps:

\ce{Br2}

reacts with D to give only E, and E reacts with

\ce{At2}

to give BrAt and D; the mass spectrum of E shows two major peaks of nearly equal intensity at m/z 205.82 and 207.82. Write the formulae of D and E. (g) 3.60 g of BrAt is synthesised; the half-life of this At isotope is 433 minutes. How long (to the nearest hour) until only 3.65 mg of BrAt remains?

Step 5 of 5: Exponential decay

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t = \frac{433\ \text{min}}{\ln 2} \times \ln\!\frac{3600}{3.65} = 433\ \text{min} \times \frac{\ln 986.3}{\ln 2} \approx 4308\ \text{min} \approx 72\ \text{h}

Analysis

$m/m_0 = 3.65/3600 = 1/986.3 = (1/2)^{t/t_{1/2}}$ , so $t = 433 \times \log_2(986.3) = 433 \times 9.945 \approx 4307$ min ≈ 72 h.

Common pitfall. Mass ratios work directly because the molar mass of BrAt cancels; just convert 3.60 g to 3600 mg first.