Relative Atomic Mass相对原子质量

Big Idea · Scale, Proportion, and Quantity (SPQ)AP Science Practice 5 · Mathematical RoutinesAP link · Unit 1~25 min

Learning goals

  • Define relative atomic mass: compared with 1/12 of the mass of a carbon-12 atom
  • Find a relative atomic mass from an atom's actual mass, and know it is close to protons + neutrons
  • Use a weighted average to explain why chlorine's relative atomic mass is 35.5

Before you start

  • Isotopes (1.3.4)

Atoms are absurdly light

An oxygen atom has a mass of about 2.657×10−262.657 \times 10^{-26} kg, and a hydrogen atom about 1.674×10−271.674 \times 10^{-27} kg. Weighing atoms in kilograms is like weighing a grain of rice in tons: the numbers are long and awkward. So chemists use a different approach. Instead of asking how many kilograms an atom has, they ask how many times heavier it is than a chosen standard.

Relative atomic mass

By international agreement, the standard is 112\frac{1}{12} of the mass of one carbon-12 atom (the carbon atom with 6 protons and 6 neutrons). An atom’s relative atomic mass相对原子质量, written ArA_r, is its mass compared with that standard:

Ar=mass of one atommass of one carbon-12 atom×112A_r = \frac{\text{mass of one atom}}{\text{mass of one carbon-12 atom} \times \frac{1}{12}}

One carbon-12 atom has a mass of 1.993×10−261.993 \times 10^{-26} kg, so the standard is about 1.661×10−271.661 \times 10^{-27} kg. For oxygen:

Ar(O)=2.657×10−26 kg1.661×10−27 kg≈16A_r(\text{O}) = \frac{2.657 \times 10^{-26}\ \text{kg}}{1.661 \times 10^{-27}\ \text{kg}} \approx 16

Note that relative atomic mass is a ratio, so it has no unit. It certainly isn’t in grams.

Relative atomic mass ≈ protons + neutrons

Protons and neutrons each have a mass very close to the standard (1/12 of a carbon-12 atom), and electrons are light enough to ignore. So for any single atom:

relative atomic mass≈protons+neutrons=mass number\text{relative atomic mass} \approx \text{protons} + \text{neutrons} = \text{mass number}

Sodium-23 has a relative atomic mass of about 23, and oxygen-16 about 16.

So why is chlorine 35.5?

The periodic table lists chlorine at 35.5 (each box on the table shows the element’s relative atomic mass; see the lookup note in lesson 1.3.3, and lesson 1.4.2 for the table itself), yet no chlorine atom has a mass number of 35.5. The reason: natural chlorine is a mixture of two isotopes, about 75.8% X35X2235Cl\ce{^{35}Cl} and about 24.2% X37X2237Cl\ce{^{37}Cl}. An element’s relative atomic mass is the weighted average of its isotopes, based on how abundant each one is.

🔧 Math tool card · Weighted AveragesSkip it if you already know this

A simple average treats every value equally. A weighted average counts each value according to its share (its “weight”):

weighted average=value1×share1+value2×share2+⋯\text{weighted average} = \text{value}_1 \times \text{share}_1 + \text{value}_2 \times \text{share}_2 + \cdots

The shares add up to 100%.

Example: in a class, homework counts for 30% of the final grade and the final exam for 70%. A student scores 90 on homework and 80 on the exam. The final grade is 90 × 30% + 80 × 70% = 27 + 56 = 83, not (90 + 80) ÷ 2 = 85. The exam carries more weight, so the grade lands closer to 80.

In a bag of candy, 60% of the pieces weigh 10 g each and 40% weigh 15 g each. What is the average mass of a piece?

The weighted average for chlorine, using mass numbers as the isotope masses:

35×75.8%+37×24.2%=26.53+8.954≈35.535 \times 75.8\% + 37 \times 24.2\% = 26.53 + 8.954 \approx 35.5

Drag the slider to change the abundances and watch which way the average moves. Then try other elements:

Try it: drag the slider to change the isotope abundances and watch the average

Example 1: From actual mass to relative atomic mass

One carbon-12 atom has a mass of 1.993×10−261.993 \times 10^{-26} kg, and one magnesium-24 atom about 3.986×10−263.986 \times 10^{-26} kg. Find the relative atomic mass of magnesium-24.

Step-by-step solution
  1. Write the standard. Standard = mass of carbon-12 × 112\frac{1}{12} = 1.993×10−26 kg12\dfrac{1.993 \times 10^{-26}\ \text{kg}}{12}.
  2. Set up. Ar=3.986×10−26 kg1.993×10−26 kg÷12=3.9861.993×12A_r = \dfrac{3.986 \times 10^{-26}\ \text{kg}}{1.993 \times 10^{-26}\ \text{kg} \div 12} = \dfrac{3.986}{1.993} \times 12.
  3. Calculate. 3.9861.993=2\dfrac{3.986}{1.993} = 2, so Ar=2×12=24A_r = 2 \times 12 = 24.
  4. Check. Magnesium-24 has 12 protons and 12 neutrons, mass number 24, which matches. The kilograms cancel, so the answer has no unit.

Example 2: Boron

Natural boron is about 20% X10X2210B\ce{^{10}B} and about 80% X11X2211B\ce{^{11}B}. Estimate boron’s relative atomic mass.

Step-by-step solution
  1. Set up the weighted average. 10×20%+11×80%10 \times 20\% + 11 \times 80\%.
  2. Calculate. 2+8.8=10.82 + 8.8 = 10.8.
  3. Check. The answer lies between 10 and 11 and closer to the more abundant isotope, 11, so it makes sense. The periodic table lists boron at 10.81.

Common mistakes

Mistake 1: “Relative atomic mass is the actual mass of an atom.” It’s a ratio compared with a standard, so it has no unit. Oxygen’s relative atomic mass is 16, not 16 g or 16 kg.

Mistake 2: Averaging the two mass numbers. (35 + 37) ÷ 2 = 36 ignores how much of each isotope there is. You need the weighted average, which gives 35.5.

Mistake 3: “Relative atomic masses are always whole numbers.” A single atom’s relative mass is close to a whole number (its mass number), but an element’s value is a weighted average of its isotopes and often isn’t, like chlorine’s 35.5.

Quick check

What is the standard for relative atomic mass?

Natural copper is about 69% X63X2263Cu\ce{^{63}Cu} and 31% X65X2265Cu\ce{^{65}Cu}. What is copper's relative atomic mass, approximately?

Summary

  • Relative atomic mass = mass of one atom ÷ (mass of a carbon-12 atom × 1/12). It’s a ratio with no unit.
  • For a single atom, the relative atomic mass is about protons + neutrons (the mass number).
  • An element’s relative atomic mass is the weighted average of its isotopes by abundance, so it often isn’t a whole number. Chlorine’s is about 35.5.
  • Weighted average: multiply each value by its share and add. The result sits closer to the value with the bigger share.
On to AP (for later)

US textbooks and the AP exam usually say average atomic mass and give it in atomic mass units, u (or amu). Since 1 u is defined as 1/12 the mass of a carbon-12 atom, the numbers are exactly the same as relative atomic masses. In topic 1.2 you’ll read isotope masses and abundances from a mass spectrum and calculate the weighted average, and topic 1.1 shows that the same number is the mass of one mole of atoms in grams (g/mol).

Practice this chapter: Module 3 · The Particle World: Atoms and Molecules · 10 questions

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