Relative Atomic Mass and Isotope Abundance Calculations

Relative Atomic Mass

• Definition: The weighted average mass of naturally occurring isotopes of an element on a scale where an atom of Carbon-12 has a mass of exactly 12 units.

Rubidium (Rb) Example

• Two isotopes of Rubidium are considered: $^{85}Rb$ and $^{87}Rb$.
• $^{85}Rb$ has a mass number of 85 and an isotopic mass of 84.91.
• $^{87}Rb$ has a mass number of 87 and an isotopic mass of 86.91.
• The relative atomic mass of Rubidium is given as 85.47.

Calculation of Abundance

• Let 'x' be the abundance of $^{85}Rb$.
• Then, the abundance of $^{87}Rb$ is (100 - x).
• The equation to calculate 'x' based on the weighted average is:
  $85.47 = (84.91 * x + 86.91 * (100 - x)) / 100$
• Simplifying the equation:
  $85.47 = (84.91x + 8691 - 86.91x) / 100$
  $8547 = 84.91x + 8691 - 86.91x$
  $-144 = -2x$
• Solving for 'x':
  $x = 72$
• Therefore, the abundance of $^{85}Rb$ is 72%.
• The abundance of $^{87}Rb$ is: 100 - 72 = 28%.

Chromium (Cr) Example

• Several isotopes of Chromium are considered:
  • $^{50}Cr$: Mass number 50, Abundance 4.35%.
  • $^{52}Cr$: Mass number 52, Abundance 83.79%.
  • $^{53}Cr$: Mass number 53, Abundance 9.50%.
  • $^{54}Cr$: Mass number 54, Abundance 2.36%.

Calculation of Relative Atomic Mass

• The relative atomic mass of Chromium is calculated using the weighted average of its isotopes:
  $Relative Atomic Mass = (50 * 4.35 + 52 * 83.79 + 53 * 9.50 + 54 * 2.36) / 100$

• Calculating the values:
  $(50 * 4.35) = 217.5$
  $(52 * 83.79) = 4357.08$
  $(53 * 9.50) = 503.5$
  $(54 * 2.36) = 127.44$

• Summing the values and dividing by 100:
  $Relative Atomic Mass = (217.5 + 4357.08 + 503.5 + 127.44) / 100$
  $Relative Atomic Mass = 5205.52 / 100$
  $Relative Atomic Mass \approx 52.0552$