To determine the half-life of protactinium-234, we can use the formula for radioactive decay:
N(t) = N₀ * (1/2)^(t / T₁/₂)
where:
N(t) is the remaining amount of the substance after time t
N₀ is the initial amount of the substance
t is the elapsed time
T₁/₂ is the half-life of the substance
In this case, we know that the initial amount N₀ is 86 g and the remaining amount N(t) after 26.76 hours is 26 g.
26 = 86 * (1/2)^(26.76 / T₁/₂)
Dividing both sides of the equation by 86:
(1/2)^(26.76 / T₁/₂) = 26/86
Taking the logarithm of both sides (base 1/2):
log(1/2)^(26.76 / T₁/₂) = log(26/86)
Using the logarithmic property: logₐ(b^c) = c * logₐ(b):
(26.76 / T₁/₂) * log(1/2) = log(26/86)
Rearranging the equation:
T₁/₂ = (26.76 * log(1/2)) / log(26/86)
Using the logarithmic properties: log(1/2) = -log(2) and log(26/86) = log(26) - log(86):
T₁/₂ = (26.76 * (-log(2))) / (log(26) - log(86))
Calculating the value:
T₁/₂ ≈ 26.76 * 0.6931 / (1.4150 - 1.9345)
T₁/₂ ≈ 18.54 hours
Therefore, the half-life of protactinium-234 is approximately 18.54 hours.