Answer :
When the rock was discovered, it had approximately 4.375 grams of europium-151.
To determine the time since the rock reached its closure temperature, we can use the concept of radioactive decay and the relationship between the half-life and the amount of remaining substance.
The equation for radioactive decay is given by:
N(t) = N₀ * (1/2)^(t / t₁/₂)
Where N(t) represents the remaining amount of substance at time t, N₀ is the initial amount of substance, t₁/₂ is the half-life, and t is the time that has elapsed.
Initial amount of samarium-151 (N₀) = 5 grams
Remaining amount of samarium-151 (N(t)) = 0.625 grams
Half-life of samarium-151 (t₁/₂) = 96.6 years
We can rearrange the equation to solve for time (t):
t = (t₁/₂) * log(N(t) / N₀) / log(1/2)
Substituting the values into the equation:
t = (96.6 years) * log(0.625 grams / 5 grams) / log(1/2)
Calculating this expression:
t ≈ (96.6 years) * (-0.3010) / (-0.6931) ≈ 41.93 years
Therefore, the time since the rock reached its closure temperature is approximately 41.93 years.
To calculate the grams of europium-151 when the rock was discovered, we can use the fact that europium-151 is the decay product of samarium-151. Since the remaining amount of samarium-151 is 0.625 grams, the grams of europium-151 will be the difference between the initial amount and the remaining amount of samarium-151:
Grams of europium-151 = 5 grams - 0.625 grams = 4.375 grams
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