What is the half-life of 44Ti if a 1.0 ng sample decays initially at a rate of 4.8 multiplied by 103 disintegrations per second?

Answer 1

Decay rate = kN

1. Convert ng to g:

1.0ng = 1.0 x 10^-9g

2. Convert g to atoms

N = (1.0x10^9 g)(1mol 44Ti/44g)(6.022x10^23atoms/1mol) = 1.37x10^13 atoms

3. Use the eqn at the top to find k

k = decay rate/N = (4.8x10^3dps)/(1.37x10^13 atoms) = 3.51x10^-10 s^-1

4. Use Half-Life eqn to find t1/2

t1/2 = 0.693/k

t1/2 = 0.693/(3.51x10^-10 s^-1) = 1.98x10^9 s

5. Convert seconds to years

1.98x10^9 s/60 = 3.30x10^7 min/60 = 5.50x10^5 hours/24 = 2.29x10^4 days/365 = 62.78 years ~ 63 years

Answer 2

The decay constant λ can be calculated from the initial decay rate: λ = decay rate / initial amount. From λ, the half-life can be calculated using the formula: half-life = ln(2) / λ. In this case, λ = 4.8 x 10^3 disintegrations/s / 1.0 ng = 4.8 x 10^3 s^-1 ng^-1, so the half-life would be approximately ln(2) / (4.8 x 10^3) ≈ 0.000144 s.