13.23 In a periodic table the average atomic mass of magnesium is given as 24.312 u. The average value is based on their relative natural abundance on earth. The three isotopes and their masses are ${}_{12}{}^{24}\mathit{M}\mathit{g}$ (23.98504u), ${}_{12}{}^{25}\mathit{M}\mathit{g}$

(24.98584u) and ${}_{12}{}^{26}\mathit{M}\mathit{g}$ (25.98259u). The natural abundance of ${}_{12}{}^{24}\mathit{M}\mathit{g}$ is 78.99% by mass. Calculate the abundances of other two isotopes.

<p><strong>13.23 </strong>Average atomic mass of magnesium, m = 24.312 u</p><p>Mass of magnesium isotope&nbsp;<span title="Click to copy mathml"><math><mmultiscripts><mrow><mrow><mi mathvariant="bold-italic">M</mi><mi mathvariant="bold-italic">g</mi></mrow></mrow><mprescripts></mprescripts><mrow><mrow><mn>12</mn></mrow></mrow><mrow><mrow><mn>24</mn></mrow></mrow></mmultiscripts></math></span> ,<span title="Click to copy mathml"><math><mprescripts></mprescripts><mrow><mrow><mn><math><msub><mrow><mrow><mi>m</mi></mrow></mrow><mrow><mrow><mn>1</mn></mrow></mrow></msub></math></mn></mrow></mrow></math></span>= 23.98504u</p><p>Mass of magnesium isotope &nbsp;<span title="Click to copy mathml"><math><mmultiscripts><mrow><mrow><mi>M</mi><mi>g</mi></mrow></mrow><mprescripts></mprescripts><mrow><mrow><mn>12</mn></mrow></mrow><mrow><mrow><mn>25</mn></mrow></mrow></mmultiscripts></math></span> ,<span title="Click to copy mathml"><math><msub><mrow><mrow><mi>m</mi></mrow></mrow><mrow><mrow><mn>2</mn></mrow></mrow></msub></math></span>= 24.98584u</p><p>Mass of magnesium isotope &nbsp;<span title="Click to copy mathml"><math><mmultiscripts><mrow><mrow><mi>M</mi><mi>g</mi></mrow></mrow><mprescripts></mprescripts><mrow><mrow><mn>12</mn></mrow></mrow><mrow><mrow><mn>26</mn></mrow></mrow></mmultiscripts></math></span> ,<span title="Click to copy mathml"><math><msub><mrow><mrow><mi>m</mi></mrow></mrow><mrow><mrow><mn>3</mn></mrow></mrow></msub></math></span>= 25.98259u</p><p>Let the abundance of magnesium isotope &nbsp;<span title="Click to copy mathml"><math><mmultiscripts><mrow><mrow><mi>M</mi><mi>g</mi></mrow></mrow><mprescripts></mprescripts><mrow><mrow><mn>12</mn></mrow></mrow><mrow><mrow><mn>24</mn></mrow></mrow></mmultiscripts></math></span> be <span title="Click to copy mathml"><math><msub><mrow><mrow><mi>η</mi></mrow></mrow><mrow><mrow><mn>1</mn></mrow></mrow></msub></math></span> = 78.99 %</p><p>Let the abundance of magnesium isotope &nbsp;<span title="Click to copy mathml"><math><mmultiscripts><mrow><mrow><mi>M</mi><mi>g</mi></mrow></mrow><mprescripts></mprescripts><mrow><mrow><mn>12</mn></mrow></mrow><mrow><mrow><mn>25</mn></mrow></mrow></mmultiscripts></math></span> be <span title="Click to copy mathml"><math><msub><mrow><mrow><mi>η</mi></mrow></mrow><mrow><mrow><mn>2</mn></mrow></mrow></msub></math></span>= x %</p><p>Therefore, the abundance of magnesium isotope &nbsp;<span title="Click to copy mathml"><math><mmultiscripts><mrow><mrow><mi>M</mi><mi>g</mi></mrow></mrow><mprescripts></mprescripts><mrow><mrow><mn>12</mn></mrow></mrow><mrow><mrow><mn>26</mn></mrow></mrow></mmultiscripts></math></span> be <span title="Click to copy mathml"><math><msub><mrow><mrow><mi>η</mi></mrow></mrow><mrow><mrow><mn>3</mn></mrow></mrow></msub></math></span>= (100 - 78.99 - x) %</p><p>= (21.01 – x)%</p><p>The average atomic mass can be expressed as:</p><p>m = <span title="Click to copy mathml"><math><mfrac><mrow><mrow><msub><mrow><mrow><mi>m</mi></mrow></mrow><mrow><mrow><mn>1</mn></mrow></mrow></msub><msub><mrow><mrow><mi>η</mi></mrow></mrow><mrow><mrow><mn>1</mn></mrow></mrow></msub><mo>+</mo><mi></mi><msub><mrow><mrow><mi>m</mi></mrow></mrow><mrow><mrow><mn>2</mn></mrow></mrow></msub><msub><mrow><mrow><mi>η</mi></mrow></mrow><mrow><mrow><mn>2</mn></mrow></mrow></msub><mo>+</mo><mi></mi><mi></mi><msub><mrow><mrow><mi>m</mi></mrow></mrow><mrow><mrow><mn>3</mn></mrow></mrow></msub><msub><mrow><mrow><mi>η</mi></mrow></mrow><mrow><mrow><mn>3</mn></mrow></mrow></msub></mrow></mrow><mrow><mrow><msub><mrow><mrow><mi>η</mi></mrow></mrow><mrow><mrow><mn>1</mn></mrow></mrow></msub><mo>+</mo><mi></mi><msub><mrow><mrow><mi>η</mi></mrow></mrow><mrow><mrow><mn>2</mn></mrow></mrow></msub><mo>+</mo><mi></mi><msub><mrow><mrow><mi>η</mi></mrow></mrow><mrow><mrow><mn>3</mn></mrow></mrow></msub></mrow></mrow></mfrac></math></span>&nbsp;=&nbsp;<span title="Click to copy mathml"><math><mfrac><mrow><mrow><mn>23.98504</mn><mo>×</mo><mn>78.99</mn><mo>+</mo><mi></mi><mn>24.98584</mn><mi>x</mi><mo>+</mo><mi></mi><mi></mi><mn>25.98259</mn><mo>(</mo><mn>21.01</mn><mo>-</mo><mi>x</mi><mo>)</mo></mrow></mrow><mrow><mrow><mn>100</mn></mrow></mrow></mfrac></math></span>&nbsp;=&nbsp;<span title="Click to copy mathml"><math><mfrac><mrow><mrow><mn>1894.578</mn><mo>+</mo><mn>24.98584</mn><mi>x</mi><mo>+</mo><mn>545.894</mn><mo>-</mo><mn>25.98259</mn><mi>x</mi></mrow></mrow><mrow><mrow><mn>100</mn></mrow></mrow></mfrac></math></span>=<span title="Click to copy mathml"><math><mfrac><mrow><mrow><mn>2440.472</mn><mo>-</mo><mn>0.99675</mn><mi>x</mi></mrow></mrow><mrow><mrow><mn>100</mn></mrow></mrow></mfrac></math></span></p><p>24.312 = <span title="Click to copy mathml"><math><mfrac><mrow><mrow><mn>2440.472</mn><mo>-</mo><mn>0.99675</mn><mi>x</mi></mrow></mrow><mrow><mrow><mn>100</mn></mrow></mrow></mfrac></math></span></p><p>x = 9.3%</p><p>Therefore the abundance of <span contenteditable="false"> <math> <mmultiscripts> <mrow> <mrow> <mi>M</mi> <mi>g</mi> </mrow> </mrow> <mprescripts></mprescripts> <mrow> <mrow> <mn>12</mn> </mrow> </mrow> <mrow> <mrow> <mn>25</mn> </mrow> </mrow> </mmultiscripts> </math> </span>is 9.3% and the abundance of&nbsp;<span title="Click to copy mathml"><math><mmultiscripts><mrow><mrow><mi>M</mi><mi>g</mi></mrow></mrow><mprescripts></mprescripts><mrow><mrow><mn>12</mn></mrow></mrow><mrow><mrow><mn>26</mn></mrow></mrow></mmultiscripts></math></span>&nbsp;is (21.01 – 9.3)11.71%</p>

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