Phosphorus-32 is a radioactive beta-emitter used in the treatment of polycythemia vera (excess red blood cells). The radioactive decay profile for the number of ³²P atoms in a 2.0-mL sample was plotted, as shown below. Based on the graph, what is the concentration of ³²P in the sample after 1.5 half-lives have passed?

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11ecba2d_123b_6675_a3bf_171426928d53_MD0008_00 In a radioactive decay, the half-life is the time required for an amount of a given isotope to decrease by half. With each subsequent half-life interval that passes, the amount remaining again decreases by half and the fraction of the initial amount that remains becomes exponentially smaller, eventually approaching zero. Accordingly, the fraction remaining after n half-lives can be calculated: 11ecba2d_123b_6676_a3bf_23e608186256_MD0008_00 Following a second half-life interval (n = 2), the fraction remaining is again cut in half, making the resulting amount equal to one fourth the initial amount. Three half-lives (n = 3) would leave an amount equal to one eighth the initial amount. In the graph of the radioactive decay of ³²P, the sample initially (0 days) has 9.0 mmol of ³²P atoms. The time required for this amount to decrease by 50% to 4.5 mmol is 14 days; therefore, the half-life of ³²P is 14 days. As such, a time interval of 1.5 half-lives is equal to 21 days. 11ecba2d_123b_6677_a3bf_83195ddc46b7_MD0008_00 At 21 days (1.5 half-lives), the graph shows that there are 3.2 mmol of ³²P atoms in the sample. To determine the molar concentration (mol/L) of ³²P in the sample, the number of moles should be divided by the 2.0-mL sample volume: 11ecba2d_123b_8d88_a3bf_7fddef504b44_MD0008_00 Therefore, the concentration of ³²P in the sample after 1.5 half-lives is 1.6 M. (Choice A) The solution has a ³²P concentration of 1.2 M after 2.0 half-lives have passed. (Choice C) The solution has a ³²P concentration of 2.3 M after 1.0 half-life has passed. (Choice D) The solution has a ³²P concentration of 3.2 M after 0.5 half-life has passed. Educational objective: A half-life is the time required for an amount of a given radioactive isotope to decrease by half. With each subsequent half-life interval that passes, the amount that remains decreases by half again, becoming exponentially smaller and eventually approaching zero. The molar concentration of a solution at a given time is determined by dividing the number of moles of analyte by the sample volume (in liters).

Phosphorus-32 Is a Radioactive Beta-Emitter Used in the Treatment of Polycythemia