60.0 grams of Th-210 with a half life of 50,000 years is allowed to decay for 250,000 years. How much of the original quantity remains? In addition, Th-210 decays via beta emission. Write the equation for its transmutation.
The long way.
k = 0.693/t1/2
ln(No/N) = kt
No = 60
N = solve for N.
k from above.
t = 250,000
First number is atomic number. Second number is mass number.
90Th210 ==> -1e0 + 91Pa210
To determine how much of the original quantity of Th-210 remains after 250,000 years, we need to use the radioactive decay formula:
N(t) = N₀ * (1/2)^(t / T₁/₂)
Where:
N₀ is the initial quantity
N(t) is the quantity remaining after time t
T₁/₂ is the half-life of the substance
In this case:
N₀ = 60.0 grams
t = 250,000 years
T₁/₂ = 50,000 years
Substituting the values into the formula:
N(t) = 60.0 g * (1/2)^(250,000 years / 50,000 years)
N(t) = 60.0 g * (1/2)^5
N(t) = 60.0 g * 1/32
N(t) = 1.875 g
Therefore, after 250,000 years, approximately 1.875 grams of the original 60.0 grams of Th-210 remains.
Th-210 decays via beta emission, which can be represented by the following equation:
Th-210 -> Pa-210 + e^-
Where:
Th-210 is the parent isotope
Pa-210 is the daughter isotope
e^- represents a beta particle (electron)
The beta particle (e^-) is emitted during the decay process of Th-210, resulting in the formation of Pa-210.