Answer :
The time since the rock reached its closure temperature is 289.8 years.
When the rock was discovered, it would contain 4.375 g of Europium 151.
If the number of atoms of samarium 151 at its closure temperature is N₀ and if it had N atoms when discovered, then,
[tex]\frac{N}{N_0} =\frac{1}{2^n}[/tex]
here, n is the number of half lives.
The mass of the isotope is proportional to the number of its atoms present in the sample. hence substitute 0.625 g for N and 5 g for N₀.
[tex]\frac{N}{N_0} =\frac{1}{2^n}\\ \frac{0.625g}{5 g} =\frac{1}{8} =\frac{1}{2^3}[/tex]
hence, the number of half live n is equal to 3.
If the half life of the isotope Samarium is 96.6 years, then the time that has elapsed from time it reaches its closure temperature and its discovery is given by,
[tex]t=3*96.6 years=289.8 years.[/tex]
Since Samarium decays to Europium, the mass of Europium at the time of discovery is equal to the mass of Samarium that has decayed.
The mass of Europium 151 at the time of discovery is given by,
[tex]m=(5g)-(0.625g)=4.375g[/tex]
Thus,the time since the rock reached its closure temperature is 289.8 years and when the rock was discovered, it would contain 4.375 g of Europium 151.
Final answer:
The rock reached its closure temperature approximately 289.8 years ago. When the rock was discovered, it had 4.375 grams of europium-151.
Explanation:
The isotope samarium-151 decays into europium-151 through a process known as radioactive decay, with a half-life of around 96.6 years. If a rock contained 5 grams of samarium-151 when it reached its closure temperature, and was found to contain 0.625 grams of samarium-151 when it was discovered, this indicates multiple half-lives have passed.
To find out how many half-lives have passed, we need to divide the original amount of samarium-151 by the final amount. In this case, 5 grams divided by 0.625 grams equals 8, which means that 3 half-lives (since 2^3 equals 8) have passed since the rock reached its closure temperature. Hence the time since the rock reached its closure temperature is approximately 3*96.6 or 289.8 years.
Every time a samarium-151 atom decays, it turns into a europium-151 atom. Therefore, the amount of europium-151 found in a rock is equal to the original amount of samarium-151 minus the remaining amount of samarium-151. In this case, 5 grams minus 0.625 grams gives 4.375 grams of europium-151 when the rock was discovered.
Learn more about Radioactive Decay here:
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