Question

Copper (Cu) consists of two naturally occurring isotopes with masses of 62.9296 and 64.9278 u. (a) How many protons and neutrons are in the nucleus of each isotope? Write the complete atomic symbol for each, showing the atomic number and mass number. (b) The average atomic mass of Cu is \(63.55 \mathrm{u}\). Calculate the abundance of each isotope.

Short answer

Cu's isotopes have atomic symbols \(^{63}_{29}\text{Cu}\) and \(^{65}_{29}\text{Cu}\). Abundances are 68.98% and 31.02%, respectively.

Step-by-step solution

Determine Atomic Number and Neutrons for Isotopes

The atomic number of copper (Cu) is 29, which means each isotope of copper has 29 protons. To find the number of neutrons in each isotope, subtract the number of protons from the mass number (rounded atomic mass).- For the isotope with mass 62.9296 u, the mass number is 63. So the number of neutrons is \(63 - 29 = 34\).- For the isotope with mass 64.9278 u, the mass number is 65. So the number of neutrons is \(65 - 29 = 36\).

Write Atomic Symbols for Each Isotope

The atomic symbol is written as \(^{A}_{Z}{X}\), where \(A\) is the mass number, \(Z\) is the atomic number, and \(X\) is the element symbol.- Isotope 1: \(^{63}_{29}\text{Cu}\)- Isotope 2: \(^{65}_{29}\text{Cu}\)

Use Average Atomic Mass Formula

The formula for the average atomic mass is given by:\[\bar{m} = (m_1 \cdot x_1) + (m_2 \cdot x_2)\]where \(\bar{m}\) is the average atomic mass, \(m_1\) and \(m_2\) are the masses of the isotopes, and \(x_1\) and \(x_2\) are the fractional abundances. Since \(x_1 + x_2 = 1\), you can express one abundance in terms of the other.

Set Up Equations to Solve for Abundance

Let \(x_1\) be the abundance of the isotope with mass 62.9296 u and \(x_2\) the abundance of the isotope with mass 64.9278 u. The equations are:\[ 63.55 = 62.9296 \cdot x_1 + 64.9278 \cdot x_2 \]\[ x_1 + x_2 = 1 \]

Solve the System of Equations

Substitute \(x_2 = 1 - x_1\) into the first equation:\[ 63.55 = 62.9296 \cdot x_1 + 64.9278 \cdot (1 - x_1) \]Simplify and solve for \(x_1\):\[ 63.55 = 62.9296x_1 + 64.9278 - 64.9278x_1 \]\[ 63.55 = -1.9982x_1 + 64.9278 \]\[ 1.9982x_1 = 64.9278 - 63.55 \]\[ x_1 = \frac{1.3778}{1.9982} \approx 0.6898 \]Using \(x_2 = 1 - x_1\):\[ x_2 \approx 0.3102 \]

Calculate Percentage Abundances

Multiply the fractional abundances by 100 to convert into percentages:- Isotope 1 abundance: \(x_1 \times 100 \approx 68.98\%\)- Isotope 2 abundance: \(x_2 \times 100 \approx 31.02\%\)

Key concepts

Atomic Number

The atomic number is a fundamental property of an element that describes the number of protons in the nucleus of its atoms. Each element is defined by its atomic number, which is unique to that element. For copper (Cu), the atomic number is 29. This means that every atom of copper has 29 protons. Protons are positively charged particles, and the number of protons not only defines the type of element but also ensures the element’s position in the periodic table.

Knowing the atomic number is crucial as it affects the element's chemical behavior and its identity. The atomic number is always the same for isotopes of an element, which are atoms of the same element with different numbers of neutrons. Despite having different numbers of neutrons, isotopes retain the same atomic number because they are chemically identical elements sharing the same number of protons.

Neutron Calculation

Calculating the number of neutrons in an isotope helps in understanding isotopic variation. Neutrons are neutral particles within an atom's nucleus that help maintain nuclear stability. The number of neutrons is determined by subtracting the atomic number from the mass number.

• Mass number is the sum of protons and neutrons in an atom.
• For the isotope with a mass of 62.9296 u, rounding the mass gives a mass number of 63.
• Subtract the atomic number (29) from the mass number (63) for the first isotope: \(63 - 29 = 34\) neutrons.
• Similarly, for the isotope with a mass of 64.9278 u, the mass number is 65.
• Subtracting gives \(65 - 29 = 36\) neutrons for the second isotope.

Knowing neutron numbers provides insight into isotopic stability and natural abundance characteristics.

Average Atomic Mass

The average atomic mass of an element reflects the weighted average of the masses of its isotopes, accounting for their relative abundances. It is the mass you see on the periodic table. For copper, its average atomic mass is specified as 63.55 u, illustrating how much each isotope contributes proportionally to copper's overall atomic mass.The average is calculated using the formula:\[ \bar{m} = (m_1 \cdot x_1) + (m_2 \cdot x_2) \]where:

• \(\bar{m}\) is the average atomic mass,
• \(m_1\) and \(m_2\) are the masses of isotopes,
• \(x_1\) and \(x_2\) are their respective fractional abundances.

Using this formula helps in finding how much each isotope weighs into the average, revealing their role in defining element identity and properties in nature.

Fractional Abundance

Fractional abundance refers to the fraction of a particular isotope present in a sample compared to the total number of isotopes. It represents how common or rare an isotope is in a natural environment. To determine the fractional abundance of isotopes, we use the relationship that the sum of the fractions is equal to 1 (or 100% when expressed as percent abundance).In the copper exercise, the abundances are expressed in terms of two equations:

• First equation: \[ 63.55 = 62.9296 \cdot x_1 + 64.9278 \cdot x_2 \] Captures the weighted contributions of isotopes.
• Second equation: \[ x_1 + x_2 = 1 \] Ensures that all isotopes sum up to 100% probability.

By solving these equations, we find:

• \( x_1 \approx 0.6898 \) indicating about 68.98% abundance for the 62.9296 u isotope.
• \( x_2 \approx 0.3102 \) meaning a 31.02% abundance for the 64.9278 u isotope.

Understanding fractional abundance is crucial for calculating the contribution of each isotope to the element's overall properties, especially in scientific and industrial applications.