Question

Chromium has an atomic mass of 51.9961 u51.9961 u and consists of four isotopes, Cr50,Cr50, Cr52,Cr52, Cr53,Cr53, and Cr54.Cr54. The Cr52Cr52 isotope has a natural abundance of 83.79%83.79% and an atomic mass of 51.9405 u.51.9405 u. The Cr54Cr54 isotope has a natural abundance of 2.37%2.37% and an atomic mass of 53.9389 u.53.9389 u. The natural abundances of the Cr50Cr50 and Cr53Cr53 isotopes exist in a ratio of 0.4579:1,0.4579:1, and the Cr50Cr50 isotope has an atomic mass of 49.9460 u.49.9460 u. Determine the atomic mass of the Cr53 isotope.Cr53 isotope.

Answer 1

Answer: The atomic mass of $_{24}^{53}\textrm{Cr}$ isotope is 52.8367 amu

Explanation:

We know that:

Total percentage abundance of the isotope = 100 %

Percentage abundance of $_{24}^{50}\textrm{Cr}\text{ and }_{24}^{53}\textrm{Cr}$ isotopes = $[100-(83.79+2.37)]=13.84\%$

We are given:

Ratio of $_{24}^{50}\textrm{Cr}\text{ and }_{24}^{53}\textrm{Cr}$ isotopes = 0.4579 : 1

Percentage abundance of $_{24}^{50}\textrm{Cr}$ isotope = $\frac{0.4579}{(0.4579+1)}\times 13.84\%=4.37\%$

Percentage abundance of $_{24}^{53}\textrm{Cr}$ isotope = $\frac{1}{(0.4579+1)}\times 13.84\%=9.49\%$

Average atomic mass of an element is defined as the sum of masses of each isotope each multiplied by their natural fractional abundance.

Formula used to calculate average atomic mass follows:

$\text{Average atomic mass }=\sum_{i=1}^n\text{(Atomic mass of an isotopes)}_i\times \text{(Fractional abundance})_i$   .....(1)

Let the mass of $_{24}^{53}\textrm{Cr}$ isotope be 'x'

• For $_{24}^{50}\textrm{Cr}$ isotope:

Mass of $_{24}^{50}\textrm{Cr}$ isotope = 49.9460 amu

Percentage abundance of $_{24}^{50}\textrm{Cr}$ = 4.37 %

Fractional abundance of $_{24}^{50}\textrm{Cr}$ isotope = 0.0437

• For $_{24}^{52}\textrm{Cr}$ isotope:

Mass of $_{24}^{52}\textrm{Cr}$ isotope = 51.9405 amu

Percentage abundance of $_{24}^{52}\textrm{Cr}$ isotope = 83.79 %

Fractional abundance of $_{24}^{52}\textrm{Cr}$ isotope = 0.8379

• For $_{24}^{53}\textrm{Cr}$ isotope:

Mass of $_{24}^{53}\textrm{Cr}$ isotope = x amu

Percentage abundance of $_{24}^{53}\textrm{Cr}$ isotope = 9.49 %

Fractional abundance of $_{24}^{53}\textrm{Cr}$ isotope = 0.0949

• For $_{24}^{54}\textrm{Cr}$ isotope:

Mass of $_{24}^{54}\textrm{Cr}$ isotope = 53.9389 amu

Percentage abundance of $_{24}^{54}\textrm{Cr}$ isotope = 2.37 %

Fractional abundance of $_{24}^{54}\textrm{Cr}$ isotope = 0.0237

Average atomic mass of chromium = 51.9961 amu

Putting values in equation 1, we get:

$51.9961=[(49.9460\times 0.0437)+(51.9405\times 0.8379)+(x\times 0.0949)+(53.9389\times 0.0237)]\\\\x=52.8367amu$

Hence, the atomic mass of $_{24}^{53}\textrm{Cr}$ isotope is 52.8367 amu