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Define the total percentage abundance.

Mathematics

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Answers

2 parts
a

(100−x)%(100 - x)\%

b

64%64\%

Step-by-step solution

5 steps

Part a· 2 steps

  1. Define the total percentage abundance.

    The total percentage abundance of all isotopes of an element is 100%100\%.

  2. Express the abundance of 71Ga^{71}\text{Ga}.

    If the abundance of 69Ga^{69}\text{Ga} is x%x\%, then the abundance of 71Ga^{71}\text{Ga} is the remaining percentage.

    Abundance of 71Ga=(100−x)%\text{Abundance of } ^{71}\text{Ga} = (100 - x)\%

    The expression for the abundance of 71Ga^{71}\text{Ga} is (100−x)%\boxed{(100 - x)\%}.

Part b

64%64\%

  1. Write the formula for Relative Atomic Mass (RAM).

    The Relative Atomic Mass (RAM) is calculated using the formula:

    RAM=(mass of isotope 1×abundance of isotope 1)+(mass of isotope 2×abundance of isotope 2)100\text{RAM} = \frac{(\text{mass of isotope 1} \times \text{abundance of isotope 1}) + (\text{mass of isotope 2} \times \text{abundance of isotope 2})}{100}
  2. Substitute the given values into the formula.

    Given:
    RAM = 69.72
    Mass of 69Ga^{69}\text{Ga} = 69
    Mass of 71Ga^{71}\text{Ga} = 71
    Abundance of 69Ga^{69}\text{Ga} = x%x\%
    Abundance of 71Ga^{71}\text{Ga} = (100−x)%(100 - x)\%

    69.72=(69×x)+(71×(100−x))10069.72 = \frac{(69 \times x) + (71 \times (100 - x))}{100}
  3. Solve for xx.

    Multiply both sides by 100:

    69.72×100=69x+71(100−x)69.72 \times 100 = 69x + 71(100 - x) 6972=69x+7100−71x6972 = 69x + 7100 - 71x

    Combine like terms:

    6972=(69−71)x+71006972 = (69 - 71)x + 7100 6972=−2x+71006972 = -2x + 7100

    Rearrange the equation to solve for xx:

    2x=7100−69722x = 7100 - 6972 2x=1282x = 128 x=1282x = \frac{128}{2} x=64x = 64

    The percentage abundance of the 69Ga^{69}\text{Ga} isotope is 64%\boxed{64\%}.

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