what is the total mass of Rn-222 remaining in an original 160-mg sample of Rn-222 after 19.1 days?
5.0 mg
To answer this question, we can use the concept of half-life and radioactive decay.
The half-life of Rn-222 is 3.82 days, meaning that every 3.82 days, the amount of Rn-222 is reduced by half.
To find the remaining mass of Rn-222 after 19.1 days, we can divide the time elapsed (19.1 days) by the half-life (3.82 days) to determine the number of half-life periods that have passed.
19.1 days / 3.82 days = 5 half-life periods
Knowing that after each half-life, the mass is reduced by half, we can use the formula:
Remaining mass = Initial mass × (1/2)^(number of half-life periods)
Substituting the given values:
Remaining mass = 160 mg × (1/2)^5
Calculating:
Remaining mass = 160 mg × 0.03125
Remaining mass ≈ 5 mg
Therefore, the total mass of Rn-222 remaining in the original 160-mg sample after 19.1 days is approximately 5 mg.
To calculate the total mass of Rn-222 remaining after 19.1 days, we need to understand the concept of radioactive decay and use the radioactive decay equation.
The radioactive decay of a substance is typically described using the half-life, which is the time it takes for half of the substance to decay. The half-life of Rn-222 is 3.82 days, meaning that after 3.82 days, half of the Rn-222 will have decayed.
To calculate the remaining mass, we can use the radioactive decay equation:
N = N0 * (1/2)^(t / t1/2)
Where:
N = Final number of atoms
N0 = Initial number of atoms
t = Time elapsed
t1/2 = Half-life of the substance
In this case, we know that the initial mass of Rn-222 is 160 mg. The molar mass of Rn-222 is approximately 222 g/mol, so we can convert the mass into the number of atoms:
N0 = (mass / molar mass) * Avogadro's constant
N0 = (160 mg / 222 g/mol) * (6.022 × 10^23 atoms/mol)
Now we can substitute the values into the equation:
N = (160 mg / 222 g/mol) * (6.022 × 10^23 atoms/mol) * (1/2)^(19.1 days / 3.82 days)
Calculating this expression will give us the final number of atoms of Rn-222. To convert this back into mass, we can multiply it by the molar mass of Rn-222:
Final mass = N * molar mass
After performing the necessary calculations, we find the final mass of Rn-222 remaining in the sample.