To find the relative abundance of the isotopes Cu-63 and Cu-65, we need to utilize the atomic mass of naturally occurring copper, which is approximately 63.55 amu.
First, let’s define the variables:
- x = fraction of Cu-63
- (1 – x) = fraction of Cu-65
Now, we can express the average atomic mass of copper in terms of these fractions:
Atomic mass of copper = (mass of Cu-63 * fraction of Cu-63) + (mass of Cu-65 * fraction of Cu-65)
This can be set up as follows:
63.55 = (62.9396 * x) + (64.9278 * (1 – x))
Expanding this equation gives:
63.55 = 62.9396x + 64.9278 – 64.9278x
Combining like terms:
63.55 = (62.9396 – 64.9278)x + 64.9278
63.55 = -1.9882x + 64.9278
Now, isolate x:
-1.9882x = 63.55 – 64.9278
-1.9882x = -1.3778
x = -1.3778 / -1.9882
x ≈ 0.6933
This result indicates that the fraction of Cu-63 is approximately 0.6933, or 69.33%. To find the fraction of Cu-65, we subtract this from 1:
1 – x ≈ 1 – 0.6933 ≈ 0.3067, which corresponds to 30.67%.
In conclusion, the relative abundances of the copper isotopes are about 69.33% Cu-63 and 30.67% Cu-65.