The molar mass of copper is 63.546. We know there are two naturally occurring isotopes: Cu-63 has a mass of 62.9296 and Cu-65 has a mass of 64.9278. Find the percentage of each isotope present naturally. Show your calculations.
The correct answer and explanation is:
To find the percentage of each isotope of copper present naturally, we use the average atomic mass formula:
Let
- xx = fraction of Cu-63
- 1−x1 – x = fraction of Cu-65
Given:
- Atomic mass of Cu = 63.546
- Mass of Cu-63 = 62.9296
- Mass of Cu-65 = 64.9278
Step 1: Set up the equation
(62.9296)(x)+(64.9278)(1−x)=63.546(62.9296)(x) + (64.9278)(1 – x) = 63.546
Step 2: Distribute and simplify
62.9296x+64.9278−64.9278x=63.54662.9296x + 64.9278 – 64.9278x = 63.546 (62.9296−64.9278)x+64.9278=63.546(62.9296 – 64.9278)x + 64.9278 = 63.546 −1.9982x+64.9278=63.546-1.9982x + 64.9278 = 63.546
Step 3: Solve for xx
−1.9982x=63.546−64.9278=−1.3818-1.9982x = 63.546 – 64.9278 = -1.3818 x=−1.3818−1.9982=0.6915x = \frac{-1.3818}{-1.9982} = 0.6915
Step 4: Convert to percentages
- Cu-63: 0.6915×100=69.15%0.6915 \times 100 = 69.15\%
- Cu-65: (1−0.6915)×100=30.85%(1 – 0.6915) \times 100 = 30.85\%
Final Answer:
- Cu-63: 69.15%
- Cu-65: 30.85%
Explanation:
The atomic mass of an element listed on the periodic table is the weighted average of all naturally occurring isotopes of that element. This value reflects both the mass and the relative abundance of each isotope. For copper, the two stable isotopes are Cu-63 and Cu-65. Their masses are slightly different due to the number of neutrons.
To find out what percentage of each isotope occurs naturally, an equation is created based on the idea that the average atomic mass equals the sum of each isotope’s mass times its relative abundance. Solving this equation gives the proportion of each isotope. The solution shows that about 69.15% of natural copper is Cu-63 and 30.85% is Cu-65. This balance ensures the average atomic mass is consistent with the known value of 63.546.