If the average atomic mass of an element having two isotopes is 71

General
1

if the average atomic mass of an element having two isotopes is 71

2

If the average atomic mass of an element having two isotopes is 71, how do you determine the relative abundance and atomic masses of the isotopes?

Answer:
To determine the relative abundance and atomic masses of the isotopes of an element with an average atomic mass of 71, you can follow these steps:

  1. Define Variables:

    • Let the atomic masses of the two isotopes be ( A ) and ( B ).
    • Let the relative abundances of the two isotopes be ( x ) and ( 1 - x ) respectively, where ( x ) is the fraction of isotope ( A ) and ( 1 - x ) is the fraction of isotope ( B ).

  2. Set Up the Equation for Average Atomic Mass:
    The average atomic mass (( \text{Avg} )) is given by the weighted sum of the atomic masses of the isotopes:
    [
    \text{Avg} = A \cdot x + B \cdot (1 - x)
    ]
    Given that the average atomic mass is 71, the equation becomes:
    [
    71 = A \cdot x + B \cdot (1 - x)
    ]

  3. Solve the Equation:
    To solve for ( x ), you need either the atomic masses ( A ) and ( B ) or additional information about the isotopes. Here are two scenarios:

    Scenario 1: Known Atomic Masses

    • Suppose you know the atomic masses of the isotopes. For example, let ( A = 70 ) and ( B = 72 ).
    • Substitute these values into the equation:
      71 = 70 \cdot x + 72 \cdot (1 - x)
    • Simplify the equation:
      71 = 70x + 72 - 72x
      [
      71 = 72 - 2x
      ]
      2x = 72 - 71
      2x = 1
      x = \frac{1}{2} = 0.5
    • Therefore, the relative abundances are:
      x = 0.5 \quad \text{(or 50% for isotope A)}
      1 - x = 0.5 \quad \text{(or 50% for isotope B)}

    Scenario 2: Unknown Atomic Masses

    • If the atomic masses ( A ) and ( B ) are unknown, additional information is needed (e.g., the mass of one isotope and the average atomic mass).

  4. General Solution with One Known Isotope Mass:

    • Suppose you know one isotope’s mass, say ( A = 70 ), and you need to find the mass of the second isotope ( B ) and their relative abundances.
    • Use the equation:
      71 = 70x + B(1 - x)
    • Rearrange to solve for ( B ):
      71 = 70x + B - Bx
      71 = B - x(B - 70)
    • If you know ( x ), you can solve for ( B ). Alternatively, if you know ( B ), you can solve for ( x ).

In summary, to determine the relative abundances and atomic masses of isotopes given an average atomic mass, you need either the atomic masses of the isotopes or additional information about their relative abundances. This process involves setting up and solving a weighted average equation.