The half life of U238 is 4.5 x 10^9 yr. A sample of rock of 1.6 g produces 29 dis/sec. Assuming all radioactivity is due to U238 find percent by mass of U 238.
I have tried
ln(t)/ln(0)= -kt and still cannot get the answer
What I get is
A) 4.5x10^9 x365x60x60x24=1.419x10^17 for half life
B) .693/1.419x10^17=4.883x10-18
C) ln t/ln 29= -(4.883x10^-18)
That is the farthest I've gotten. Can someone help?
Use Nt/N0 = e^{-kt}
1[yr] = a = 31557600[s]
(1/2) = e^(-kh)
Where:
Half life: h= 4.5e9 * a = 1.420092E+17[s]
k = -ln(0.5)/1.420092E+17
.: k = 4.8810019E-18[/s]
Okay, so far. To continue...
Assume there is n atoms of U238 in the sample. t=1 second latter there is n-29. (via 29 disintegrations per second)
(n-29)/n = e^{-kt}
1 - (29/n) = e^{-kt}
29/n = 1-e^{-kt}
n = 29/(1-e^{-kt})
n = 29/(1-exp(-4.8810019E-18))
n = 5.9414031E+18[atoms]
Next: find the weight of this number.
Given:
1[mole]=6.02214129(27)E+23[atoms]
U[238] weighs 238.05078822[g/mol]
Express as a percentage of the sample weight (1.6[g]).
To find the percent by mass of U238 in the rock sample, we need to use the equation for radioactive decay:
N(t) = N0 * e^(-λt)
where:
N(t) is the number of radioactive atoms at time t
N0 is the initial number of radioactive atoms
λ is the decay constant
t is the time in seconds
Given that the rock sample produces 29 disintegrations per second (dis/sec) and assuming all radioactivity is due to U238, we can set N0 = 29 and solve for the decay constant, λ.
29 = 29 * e^(-λt)
Simplifying, we have:
1 = e^(-λt)
Take the natural logarithm of both sides:
ln(1) = ln(e^(-λt))
0 = -λt
Rearranging, we get:
λt = 0
Now, we know that the half-life of U238 is 4.5 x 10^9 years. Since 1 year is approximately equal to 3.1536 x 10^7 seconds, we can convert the half-life to seconds:
half_life_seconds = 4.5 x 10^9 years * 3.1536 x 10^7 seconds/year
half_life_seconds = 1.41984 x 10^17 seconds
Substituting this value into the equation:
λ * 1.41984 x 10^17 = 0
Therefore, λ = 0.
Since λ = 0, it means that there is no decay happening, which contradicts the information given. It seems there might be an error in the provided values or calculations. Please recheck the given information or share any additional data you might have.
To find the percent by mass of U238 in the rock sample, you need to use the concept of radioactive decay and the relationship between the half-life and decay constant.
First, let's understand the formula you mentioned: ln(t)/ln(0) = -kt.
Here, t represents time, l is the natural logarithm, and k is the decay constant. This formula is used to calculate the number of remaining radioactive atoms over time t, given the initial number of radioactive atoms.
However, to solve this problem, we need to use a slightly different equation related to the decay rate, R, which is given as the number of disintegrations per unit time.
The equation relating the decay rate, decay constant, and the number of radioactive atoms is: R = λN, where λ is the decay constant and N is the number of radioactive atoms.
Now, let's proceed to solve the problem step by step:
1. Calculate the decay constant (λ) using the half-life (t1/2) of U238:
λ = ln(2) / t1/2
From the given information, the half-life of U238 is 4.5 x 10^9 years. So,
λ = ln(2) / (4.5 x 10^9 years)
2. Calculate the number of radioactive U238 atoms (N) in the sample:
N = R / λ
From the given information, the decay rate (R) is 29 disintegrations per second. So,
N = 29 / λ
3. Calculate the mass of U238 in the sample:
Mass of U238 = (N x atomic mass of U238) / Avogadro's number
The atomic mass of U238 is 238 g/mol, and Avogadro's number is approximately 6.022 x 10^23.
4. Finally, calculate the percent by mass of U238:
Percent by mass of U238 = (Mass of U238 / Mass of the sample) x 100
By plugging in the numbers into these formulas, you should be able to find the percent by mass of U238 in the rock sample.