Four isotopes of Lead occur naturally:
204Pb (mass = 203.973037 amu)
206Pb (mass = 205.974455 amu)
207Pb (mass = 206.975885 amu)
208Pb (mass = 207.976641 amu)
If the percent abundance of 204Pb and 208Pb are 1.400 and 52.40, respectively, what is the percent abundance of 206Pb?
Pb average atomic mass = 207.2
abundance 204Pb = 1.400%
abundance 208Pb = 52.40%
abundance 206Pb = Y (as a fraction)
abundance 207Pb = Y-1.400-52.40
0.14(203.973037) + Y(205.974455) + (Y-1.400-52.50)(206.975885) + 0.5240(207.976641) = 207.2
Solve for Y
To find the percent abundance of 206Pb, we can use the information provided and the average atomic mass of Pb.
Let's denote the percent abundance of 206Pb as x.
Since the sum of the percent abundances of all isotopes should be 100%, we have the equation:
1.400 + x + 52.40 = 100
Combine the known percent abundances:
1.400 + 52.40 = 53.80
Subtract 53.80 from both sides:
x = 100 - 53.80
x = 46.20
Therefore, the percent abundance of 206Pb is 46.20%.
To determine the percent abundance of 206Pb, we need to use the concept that the average atomic mass is a weighted average of the isotopes' masses, weighted by their relative abundance.
First, we can find the mass contribution of each isotope by multiplying the percent abundance by the mass of the isotope:
Mass contribution of 204Pb = (1.400/100) * 203.973037 amu = 2.855822 amu
Mass contribution of 206Pb = (x/100) * 205.974455 amu (To be determined)
Mass contribution of 207Pb = (100 - 1.400 - 52.40) * 206.975885 amu = 7285.538735 amu
Mass contribution of 208Pb = (52.40/100) * 207.976641 amu = 108.9528584 amu
Since the average atomic mass is given as 207.2 amu, we can set up the equation:
207.2 amu = 2.855822 amu + (x/100) * 205.974455 amu + 7285.538735 amu + 108.9528584 amu
Simplifying the equation:
207.2 amu = 7397.347415 amu + (x/100) * 205.974455 amu
Rearranging the equation to solve for x, the percent abundance of 206Pb:
(x/100) * 205.974455 amu = 207.2 amu - 7397.347415 amu
(x/100) * 205.974455 amu = -7190.147415 amu
x * 205.974455 amu = (-7190.147415 amu) * 100
x * 205.974455 amu = -719014.7415 amu
x = (-719014.7415 amu) / 205.974455 amu
x ≈ -3494.440392
Since the percent abundance cannot be negative, it means there was an error in the calculations or the given information. Please double-check the values and ensure they are correct.