Question

Calculate the fraction of each of the following radionuclides that remains after 1 day, 2 days, 3 days, and 4 days (half-lives are given in parentheses): iron-59 (44.51 days), titanium-45 (3.078 h), calcium-47 (4.536 days), and phosphorus-33 (25.3 days).

Short answer

The fraction of radionuclide which will remain after 1 day, 2 days, 3 days and 4 days:

Step-by-step solution

Given information

For radioactive elements, iron-59, titanium-45, calcium-47, and phosphorous-33, the formula for fraction of sample and half time of sample is used to calculate the fraction of radionuclide that will remain after 1 day, 2 days, 3 days, and 4 days.

Concept

The first order of kinetics states:

$\ln \left(\frac{N}{N_0}\right)=-\lambda t$

Here, $N$is final concentration and $N_0$is initial concentrations, $\lambda$is the decay constant, $t$and is the passage of time.

localid="1646068104244" $t_{1/2}(\text{half}-\text{time})=ln\frac{2}{\lambda }$

The half-life time is $t_{1/2}$in this case.

Solution

The following formula can be used to compute the sample half-time:

$t_{1/2}=\ln \frac{2}{\lambda }$

The following formula can be used to compute the proportion of sample and time:

$\ln \left(\frac{N}{N_0}\right)=-\lambda t$

Where,

$t_{1/2}=$half time

$\lambda =$decay constant

$\mathrm{N}\text{and}{\mathrm{N}}_0=$the number of nuclei that are radioactive.

The radioactive elements' half-lives are stated as
days of iron-$59$= $44.51$

hours of titanium-$45$= $3.078$

days of calcium-$47$= $4.536$

days of phosphorus-$33$ = $25.3$

The above table's spreadsheet documentation is

localid="1646067528747" $\text{CellE5}=LN\left(2\right)/D5$

localid="1646067533137" $\text{CellBI1}=E5*C9*-1$

localid="1646067538088" $\text{CellC11}=EXP\left(B11\right)$

localid="1646067542914" $\text{CellEl1}=E5*F9*-1$

localid="1646067547700" $\text{CellF11}=EXP\left(E11\right)$

localid="1646067552312" $\text{CellC18}=E5*D16*-1$

localid="1646067557274" $\text{CellD18}=EXP\left(C18\right)$

localid="1646067562817" $\text{CellF18}=E5*G16*-1$

localid="1646067567937" $\text{CellG18}=EXP\left(F18\right)$

The following data will be calculated using the given formulas:

Conclusion 

After $1$day, $2$days, $3$days, and $4$days, the fraction that remained is -