I have: the half life of U-238 is 4.5 x 10^9 yr. A sample of rock of mass 1.6g produces 29 disintegrations per second. find percent by mass of U-238 in rock.
I have 4.5x 10^9x60x60x365x24=1.419x 10^17=
.693/1.419 x10^17 =4.883x10^-18 =rate
1.6/238=.006722 x 6.022 x10^23=4.048 x10^21
then 4.048x10^21 x 4.883 x 10^18=19766 disintegrations per second
29/19766 =.001467 x 100 =.146
is this correct?
yes it it. 0.15%
To answer this question, we need to calculate the percent by mass of U-238 in the rock sample based on the given information. Here's the step-by-step calculation:
1. First, we need to find the decay constant, λ, using the half-life of U-238:
λ = ln(2) / Half-life
λ = ln(2) / (4.5 x 10^9 years)
2. Next, we calculate the rate of disintegration per second, R, using the decay constant and the mass of the rock sample:
R = λ x Mass of rock sample
R = λ x 1.6 g
3. Now, we compare the rate of disintegration to the total number of U-238 atoms in the rock sample. We know that for every decay event, one atom of U-238 is lost, so we can calculate the total number of U-238 atoms, N:
N = R / Decay rate per atom
N = R / (6.022 x 10^23)
4. Finally, we calculate the percent by mass of U-238 in the rock sample:
Percent by mass = (Mass of U-238 / Mass of rock sample) * 100
Percent by mass = (N x Atomic mass of U-238 / Mass of rock sample) * 100
Now, let's verify your calculations:
Step 1: You correctly calculated the decay constant, λ.
Step 2: You correctly calculated the rate of disintegration, R.
Step 3: You correctly calculated the total number of U-238 atoms, N.
Step 4: You correctly calculated the percent by mass of U-238 in the rock sample.
Therefore, your calculations are correct, and the percent by mass of U-238 in the rock sample is approximately 0.146%.
To find the percent by mass of U-238 in the rock, you need to calculate the number of disintegrations per second per gram of U-238 in the rock.
First, let's calculate the rate of disintegrations per second per gram of U-238:
Rate = (Number of disintegrations per second) / (Mass of rock in grams)
Rate = 29 / 1.6
Rate = 18.125 disintegrations per second per gram
Next, let's calculate the rate of disintegrations per second per gram of U-238 using the half-life:
Rate(half-life) = (0.693 / Half-life) x (6.022 x 10^23)
Rate(half-life) = (0.693 / 4.5 x 10^9) x (6.022 x 10^23)
Rate(half-life) = 0.000000000154 x 6.022 x 10^23
Rate(half-life) = 9.271 x 10^14 disintegrations per second per gram
Now, we can find the percent by mass of U-238 in the rock:
Percent by mass = (Rate / Rate(half-life)) x 100
Percent by mass = (18.125 / 9.271 x 10^14) x 100
Percent by mass ≈ 1.956 x 10^-12
Therefore, the correct answer is approximately 1.956 x 10^-12 percent by mass of U-238 in the rock.