Question

The isotope 65Ga undergoes radioactive decay, with decay constant i = 0.0456/min. What is the half-life of 65Ga? min How long will it take for 60.0% of a sample of 65Ga to decay? min What is the activity (rate of decay) for 8 mg Ga-65, in decays/second? decays/s

Answer 1

The half-life of 65Ga is 15.19 minutes. It takes 12.84 minutes for 60% of a sample of 65Ga to decay. For 8 mg of 65Ga, the decay activity is 5.61 * 10^12 decays per second.

Step-by-step explanation:

Calculating the Half-Life of 65Ga

The half-life (t1/2) of an isotope is calculated using the decay constant (λ), which is given by the formula t1/2 = ln(2)/λ. Substituting the given decay constant for 65Ga, λ = 0.0456/min, the half-life is t1/2 = ln(2) / 0.0456/min, which yields:

t1/2 = 15.19 min.

Calculating the Time for 60% Decay

To find out how long it will take for 60% of a sample of 65Ga to decay, we can use the formula N = N0e−λt, where N0 is the initial amount, N is the remaining amount, and t is the time. For a 60% decay, 40% (0.4) remains, and we can solve for t:

0.4N0 = N0e−λt

t = −(ln(0.4)/λ)

t = 12.84 min.

Calculating the Activity for 8mg of 65Ga

To calculate the activity (A), we use the formula A = λN, where N is the number of atoms. First, we convert mass to moles to atoms, considering the atomic weight of 65Ga is approximately 65 g/mol:

8 mg * (1 g/1000 mg) * (1 mol/65 g) * (6.022 * 1023 atoms/mol) = 7.40 * 1018 atoms.

Then the activity is A = λ * N:

A = 0.0456/min * 7.40 * 1018 atoms * (1 min/60 s) = 5.61 * 1012 decays/s.

Answer 2

The half-life of 65Ga is 15.2 min. It will take 24.6 min for 60.0% of a sample of 65Ga to decay. The activity for 8 mg of Ga-65 is 3.36 × 10¹⁶ decays/s.

Step-by-step explanation:

The half-life of an isotope is the time it takes for half of the initial quantity of the isotope to decay. In this case, the isotope 65Ga has a decay constant of i = 0.0456/min. To find the half-life, we can use the equation:

t(1/2) = ln(2) / i

Plugging in the given value for i, we get:

t(1/2) = ln(2) / 0.0456 = 15.2 min

To find the time it will take for 60.0% of a sample of 65Ga to decay, we can use the equation:

t = (ln(1-x) / -i)

Where x is the fraction of the sample that remains. Plugging in the given value of x = 0.60 and i = 0.0456, we get:

t = (ln(1-0.60) / -0.0456) = 24.6 min

To find the activity (rate of decay) for 8 mg of Ga-65, we can use the equation:

activity = (decay constant) * (number of atoms)

We can calculate the number of atoms from the mass of Ga-65 using Avogadro's number (6.022 × 10²³ atoms/mol). Using the atomic mass of Ga-65 (64.92 g/mol), we find that 8 mg of Ga-65 is equal to 8 × 10⁻⁶ g. Plugging in these values, we get:

number of atoms = (8 × 10⁻⁶ g) / (64.92 g/mol) × (6.022 × 10²³ atoms/mol) = 7.37 × 10¹⁷ atoms

Now, we can calculate the activity:

activity = (0.0456/min) * (7.37 × 10¹⁷ atoms) = 3.36 × 10¹⁶ decays/s