Answer :
The amount after 1000 years will be, 5.19 grams.
The amount after 10000 years will be, 0.105 grams.
Step-by-step explanation :
Half-life = 1599 years
First we have to calculate the rate constant, we use the formula :
![k=\frac{0.693}{t_{1/2}}](https://tex.z-dn.net/?f=k%3D%5Cfrac%7B0.693%7D%7Bt_%7B1%2F2%7D%7D)
![k=\frac{0.693}{1599\text{ years}}](https://tex.z-dn.net/?f=k%3D%5Cfrac%7B0.693%7D%7B1599%5Ctext%7B%20years%7D%7D)
![k=4.33\times 10^{-4}\text{ years}^{-1}](https://tex.z-dn.net/?f=k%3D4.33%5Ctimes%2010%5E%7B-4%7D%5Ctext%7B%20years%7D%5E%7B-1%7D)
Now we have to calculate the amount after 1000 years.
Expression for rate law for first order kinetics is given by:
![t=\frac{2.303}{k}\log\frac{a}{a-x}](https://tex.z-dn.net/?f=t%3D%5Cfrac%7B2.303%7D%7Bk%7D%5Clog%5Cfrac%7Ba%7D%7Ba-x%7D)
where,
k = rate constant = ![4.33\times 10^{-4}\text{ years}^{-1}](https://tex.z-dn.net/?f=4.33%5Ctimes%2010%5E%7B-4%7D%5Ctext%7B%20years%7D%5E%7B-1%7D)
t = time passed by the sample = 1000 years
a = initial amount of the reactant = 8 g
a - x = amount left after decay process = ?
Now put all the given values in above equation, we get
![1000=\frac{2.303}{4.33\times 10^{-4}}\log\frac{8}{a-x}](https://tex.z-dn.net/?f=1000%3D%5Cfrac%7B2.303%7D%7B4.33%5Ctimes%2010%5E%7B-4%7D%7D%5Clog%5Cfrac%7B8%7D%7Ba-x%7D)
![a-x=5.19g](https://tex.z-dn.net/?f=a-x%3D5.19g)
Thus, the amount after 1000 years will be, 5.19 grams.
Now we have to calculate the amount after 10000 years.
Expression for rate law for first order kinetics is given by:
![t=\frac{2.303}{k}\log\frac{a}{a-x}](https://tex.z-dn.net/?f=t%3D%5Cfrac%7B2.303%7D%7Bk%7D%5Clog%5Cfrac%7Ba%7D%7Ba-x%7D)
where,
k = rate constant = ![4.33\times 10^{-4}\text{ years}^{-1}](https://tex.z-dn.net/?f=4.33%5Ctimes%2010%5E%7B-4%7D%5Ctext%7B%20years%7D%5E%7B-1%7D)
t = time passed by the sample = 10000 years
a = initial amount of the reactant = 8 g
a - x = amount left after decay process = ?
Now put all the given values in above equation, we get
![10000=\frac{2.303}{4.33\times 10^{-4}}\log\frac{8}{a-x}](https://tex.z-dn.net/?f=10000%3D%5Cfrac%7B2.303%7D%7B4.33%5Ctimes%2010%5E%7B-4%7D%7D%5Clog%5Cfrac%7B8%7D%7Ba-x%7D)
![a-x=0.105g](https://tex.z-dn.net/?f=a-x%3D0.105g)
Thus, the amount after 10000 years will be, 0.105 grams.