Potassium has an atomic mass of 39.10amu. a sample of potassium is made up of isotopes potassium-39 (mass=38.964) and potassium-41(mass =40.962 amu). What percentage of the potassium sample is potassium-39? what percent of the sample is potassium-41?
For a sample of K, with x grams of 39K and y grams of 41K the total mass will be
38.964x + 40.962y
Assuming a sample of 100g, we have
38.964x + 40.962(100-x) = 39.10(100)
x = 93.193
39K is 93.193%
41K = 6.807%
check:
38.964*.93193 + 40.962*.06807 = 39.10
To find the percentage of the potassium sample that is potassium-39 and potassium-41, we can follow these steps:
Step 1: Calculate the relative abundance of each isotope.
The relative abundance of an isotope refers to the percentage or decimal fraction of that particular isotope in a sample. The sum of the relative abundances of all isotopes in a sample should be equal to 1 or 100%.
Let x be the relative abundance of potassium-39, and (1 - x) be the relative abundance of potassium-41.
Step 2: Write down the atomic mass of each isotope.
Given:
Atomic mass of potassium-39 (mass1) = 38.964 amu
Atomic mass of potassium-41 (mass2) = 40.962 amu
Step 3: Set up an equation for the average atomic mass of the potassium sample.
The average atomic mass of a sample is calculated by multiplying the relative abundance of each isotope by its atomic mass and then summing the results.
Average atomic mass = (relative abundance of isotope 1 * atomic mass of isotope 1) + (relative abundance of isotope 2 * atomic mass of isotope 2)
39.10 amu = (x * 38.964 amu) + ((1 - x) * 40.962 amu)
Step 4: Solve the equation for x.
To find x, solve the equation derived in the previous step for x.
39.10 amu = 38.964x + 40.962 - 40.962x
39.10 - 40.962 = -1.998x
-1.862 = -1.998x
x = -1.862 / -1.998
x ≈ 0.932
Step 5: Calculate the percentages.
To find the percentage of each isotope in the sample, multiply the relative abundance by 100%.
Percentage of potassium-39 = x * 100%
Percentage of potassium-41 = (1 - x) * 100%
Percentage of potassium-39 ≈ 0.932 * 100% ≈ 93.2%
Percentage of potassium-41 ≈ (1 - 0.932) * 100% ≈ 6.8%
Therefore, approximately 93.2% of the potassium sample is potassium-39, while approximately 6.8% of the potassium sample is potassium-41.