Isotopes and Relative Atomic Mass Notes

Isotopes and Relative Atomic Mass

Learning Intentions

  • Define the term "isotope."
  • Compare isotopes based on given information.
  • Define Relative Atomic Mass (RAM).
  • Calculate RAM of isotopes given their abundance.
  • Calculate RAM to a specific number of significant figures or decimal places.

Keywords

  • Isotope: Variations of the same element, differing in mass.
  • Atomic Mass: The mass of a particular atom.
  • Relative Atomic Mass: The average mass of a sample of atoms.
  • Abundance: The quantity of something present in a sample.

Atomic Structure Review

Atoms contain protons (+, positive charge), neutrons (0, neutral charge), and electrons (-, negative charge).

  • Atoms have a fixed number of protons, determined by their atomic number (which defines the element).
  • Neutral atoms have an equal number of protons and electrons.
  • The number of neutrons can be calculated using the mass number and the number of protons.

Isotopes Explained

Isotopes are variations of the same element that have the same number of protons but different numbers of neutrons, leading to differences in mass number.

Examples of Hydrogen Isotopes:

  • Protium (Hydrogen-1): 1 proton, 0 neutrons
  • Deuterium (Hydrogen-2): 1 proton, 1 neutron
  • Tritium (Hydrogen-3): 1 proton, 2 neutrons

Drawing and Labeling Isotopes

Isotopes of an element can be represented by drawing their nucleus and indicating the number of protons and neutrons.

Examples of Carbon Isotopes:

  • Carbon-12
  • Carbon-13
  • Carbon-14

Oxygen Isotopes

Oxygen has three common isotopes:

  • Oxygen-16: 8 protons, 8 neutrons
  • Oxygen-17: 8 protons, 9 neutrons
  • Oxygen-18: 8 protons, 10 neutrons

Relative Atomic Mass (RAM)

Isotopes of an element are typically found mixed together.
The Relative Atomic Mass (RAM) is the average mass of a common sample of isotopes for that element.

Calculating RAM

The RAM is calculated by considering the relative abundance of each isotope in a sample.

For example, Hydrogen has three isotopes with the following approximate abundances:

  • Protium (1H): 99.98%
  • Deuterium (2H): 0.02%
  • Tritium (3H): ~0.00%

Calculation of RAM for Hydrogen:

  1. Convert percentages to decimals by dividing by 100.
    • Protium: \frac{99.98}{100} = 0.9998
    • Deuterium: \frac{0.02}{100} = 0.0002
    • Tritium: \frac{0}{100} = 0
  2. Multiply the decimal by the mass number of the isotope.
    • Protium: 0.9998 \times 1 = 0.9998
    • Deuterium: 0.0002 \times 2 = 0.0004
    • Tritium: 0\times 3 = 0
  3. Add the results to find the RAM.
    • 0.9998 + 0.0004 + 0 = 1.0002 \text{ uamu}

Example Calculation: Boron

Boron has two isotopes:

  • Boron-10: 19.9% abundance
  • Boron-11: 80.1% abundance

Calculation:

  1. Convert percentages to decimals:
    • Boron-10: \frac{19.9}{100} = 0.199
    • Boron-11: \frac{80.1}{100} = 0.801
  2. Multiply by the mass number:
    • Boron-10: 0.199 \times 10 = 1.99
    • Boron-11: 0.801 \times 11 = 8.811
  3. Add the results:
    • 1.99 + 8.811 = 10.801 \text{ uamu}

Accuracy in Numeric Answers

  • Answers can be given to a specific number of decimal places or significant figures.

Decimal Places

To round to a certain number of decimal places:

  1. Identify the digit in the desired decimal place.
  2. Look at the digit immediately to the right.
  3. If it is 5 or greater, add 1 to the digit in the desired decimal place.
  4. If it is less than 5, keep the digit in the desired decimal place as it is.

Example: Rounding Boron RAM to Two Decimal Places

Boron RAM calculated as 10.801 uamu.
To round to two decimal places, we look at the third decimal place (1).
Since 1 < 5, we keep the second decimal place (0) as it is.

  • Rounded RAM = 10.80 uamu

Example Calculation: Magnesium

Magnesium has three isotopes:

  • Magnesium-24: 78.20% abundance
  • Magnesium-25: 10.11% abundance
  • Magnesium-26: 11.69% abundance

Calculation:

  1. Convert percentages to decimals and multiply by mass number:
    • Magnesium-24: \frac{78.20}{100} \times 24 = 18.768
    • Magnesium-25: \frac{10.11}{100} \times 25 = 2.5275
    • Magnesium-26: \frac{11.69}{100} \times 26 = 3.0394
  2. Add the results:
    • 18.768 + 2.5275 + 3.0394 = 24.3349 \text{ uamu}
  3. Round to two decimal places: 24.33 uamu

Significant Figures Rules

  • All digits in numbers expressed in standard form are significant (e.g., 4.320 \times 10^{-6} has 4 significant figures).
  • All non-zero numbers are significant (e.g., 42.3 has 3 significant figures).
  • Zeros between two non-zero numbers are significant (e.g., 4.302 has 4 significant figures).
  • Leading zeros are not significant (e.g., 0.0043 has 2 significant figures).
  • Trailing zeros to the right of a decimal point are significant (e.g., 42.00 has 4 significant figures).
  • For numbers less than 1, 0.4 has 1 significant figure, 0.04 also has 1 significant figure, whereas 0.40 has 2 significant figures and 0.400 has 3 significant figures.
  • Whole numbers written without a decimal point will have the same number of significant figures as the number of digits, with the assumption that the decimal point occurs at the end of the number (e.g., 400 has 3 significant figures). Therefore, a measured distance of 100 m will be considered as having three significant figures.

Significant Figures Examples

  • 2.183 has 4 significant figures
  • 4.21 has 3 significant figures
  • 9.3713 has 5 significant figures
  • 20.14 has 4 significant figures
  • 0.00082 has 2 significant figures

More Significant Figures Examples

  • 3.25 has 3 significant figures
  • 16.6894 has 6 significant figures
  • 4.032 has 4 significant figures
  • 0.0043 has 2 significant figures
  • 0.09082 has 4 significant figures

Example Calculation: Magnesium (Significant Figures)

Using the previous magnesium isotope data:

  • Magnesium-24: 78.20%
  • Magnesium-25: 10.11%
  • Magnesium-26: 11.69%

Calculations yield:

  • Magnesium-24 contribution: 18.768 (5 sig figs)
  • Magnesium-25 contribution: 2.5275 (5 sig figs)
  • Magnesium-26 contribution: 3.0394 (5 sig figs)
  • Total RAM before rounding: 24.3349 uamu (6 sig figs)

Rounded RAM to 4 sig figs: 24.33 uamu

Example Calculation: Lithium (Significant Figures)

Lithium has two isotopes:

  • Lithium-6: 6.85% abundance
  • Lithium-7: 93.15% abundance

Calculation:

  1. Convert percentages to decimals and multiply by mass number:
    • Lithium-6: \frac{6.85}{100} \times 6 = 0.411
    • Lithium-7: \frac{93.15}{100} \times 7 = 6.5205
  2. Add the results: 0.411 + 6.5205 = 6.9315 \text{ uamu}
  3. Since 6.85% has 3 significant figures, round the result to 3 significant figures: 6.93 uamu

Isotope Drawing Exercises

  • Nitrogen-14: 7 protons, 7 neutrons
  • Boron-10: 5 protons, 5 neutrons
  • Beryllium-7: 4 protons, 3 neutrons

Example Calculation: Bromine

Bromine has two isotopes:

  • Bromine-79: 50.7% abundance
  • Bromine-80: 49.3% abundance

Calculation:

  1. Convert percentages to decimals and multiply by mass number:
    • Bromine-79: \frac{50.7}{100} \times 79 = 40.053
    • Bromine-80: \frac{49.3}{100} \times 80 = 39.44
  2. Add the results: 40.053 + 39.44 = 79.493 \text{ uamu}
  3. Round to one decimal place: 79.5 uamu

Example Calculation: Carbon (with 3 isotopes)

Carbon has three isotopes:

  • Carbon-12: 98% abundance
  • Carbon-13: 1.5% abundance
  • Carbon-14: 0.50% abundance

Calculation:

  1. Convert percentages to decimals and multiply by mass number:
    • Carbon-12: \frac{98}{100} \times 12 = 11.76
    • Carbon-13: \frac{1.5}{100} \times 13 = 0.195
    • Carbon-14: \frac{0.50}{100} \times 14 = 0.07
  2. Add the results: 11.76 + 0.195 + 0.07 = 12.025 \text{ uamu}
  3. Round to the least number of significant figures (2 sig figs): 12 uamu

Review

  • Defined the term "isotope."
  • Compared isotopes based on given information.
  • Defined Relative Atomic Mass (RAM).
  • Calculated RAM of isotopes given their abundance.
  • Calculated RAM to a specific number of significant figures or decimal places.