Answer:
56.4459u
Explanation:
Given parameters:
Atomic mass of Cr = 51.9961u
Isotopes listed = Cr50, Cr52, Cr53 and Cr4
Natural abundance of Cr52 = 83.79% Atomic mass = 51.9405u
Natural abundance of Cr54 = 2.37% Atomic mass = 53.9389u
Natural abundance of Cr50:Cr53 = 1:0.4579 Atomic mass of Cr50 = 49.9460u
Unknown:
Atomic mass of Cr53 = ?
Solution:
Isotopes of an element have the same electronic configuration. They differ only in their masses. Isotopes have been studied extensively using the mass spectrometry method. The method is used to determine the proportion/percentage/fraction by which each of the isotopes of an element occur in nature.
The proportion is called the geonormal abundance or simply abundance. From this proportion, the relative atomic mass(RAM) of an element can be evaluated. The RAM is the average mass of all isotopes of an element.
RAM = m₁α₁ + m₂α₂ + m₃α₃ + ........... + mₙαₙ
where m is the mass of a given isotope
α is the abundance of the isotope
To solve the problem we need to establish the relationship between the given atomic masses, their relative atomic mass and the abundances of the isotope:
RAM = m₅₂α₅₂ + m₅₀α₅₀ + m₅₃α₅₃ + m₅₄α₅₄
The unknown in the expression here is the m₅₃, atomic mass of Cr53
We need to derive the abundances α₅₃ and α₅₀.
Given that their isotopic natural abundances is 1:0.4579;
The sum of natural abundances when expressed as percentage is equal to 100:
Therefore;
natural abundance of α₅₃ and α₅₀ = 100 - (83.79+2.37) = 13.84%
Since they occur in ratio 1:0.4579, we can determine their proportions.
sum of the ratio = 1+0.4579 = 1.4579
Proportion of α₅₃ = [tex]\frac{0.4579}{1.4579}[/tex] x 13.84 = 4.3469%
Proportion of α₅₀ = [tex]\frac{1}{1.4579}[/tex] x 13.84u = 9.4931%
Since we now know the abundances of the remaining isotopes, we can forge ahead to find the unknown atomic mass of Cr53.
Evaluating
RAM = m₅₂α₅₂ + m₅₀α₅₀ + m₅₃α₅₃ + m₅₄α₅₄
51.9961 =( 51.9405 x [tex]\frac{83.79}{100}[/tex] ) + ( 49.9460 x [tex]\frac{9.4931}{100}[/tex] + (m₅₃ x [tex]\frac{4.3469}{100}[/tex] ) + (53.9389 x [tex]\frac{2.37}{100}[/tex])
51.9961 = 43.5209 + 4.7414 + 0.0435m₅₃ + 1.2784
0.0435m₅₃ = 51.9961 - 49.5407
0.0435m₅₃ = 2.4554
m₅₃ = [tex]\frac{2.4554}{0.0435}[/tex] = 56.4459u
