Question

Congratulations! You discover a new element you name wyzzlebium \((W z)\). The average atomic mass of \(W z\) was found to be \(303.001 \mathrm{u}\), and its atomic number is 120 . (a) If the masses of the two isotopes of wyzzlebium are \(300.9326 \mathrm{u}\) and \(303.9303 \mathrm{u}\), what is the relative abundance of each isotope? (b) What are the isotopic notations of the two isotopes? (e.g., \({ }_{\text {Z W W ) }}\) (c) How many neutrons are in one atom of the more abundant isotope?

Short answer

31% for Isotope 1, and 69% for Isotope 2. \({}^{301}_{120}Wz\) and \({}^{304}_{120}Wz\). 184 neutrons in the more abundant isotope.

Step-by-step solution

- Define the given information

The average atomic mass of wyzzlebium \(W z\) is 303.001 u. The masses of the two isotopes are 300.9326 u (Isotope 1) and 303.9303 u (Isotope 2). The atomic number \(Z\) is 120.

- Set up equations for relative abundance

Let the relative abundance of Isotope 1 be \( x \). Therefore, the relative abundance of Isotope 2 will be \( (1-x) \). Using the formula for average atomic mass, we set up the following equation:\[ 300.9326x + 303.9303(1 - x) = 303.001 \]

- Solve for x

Distribute and simplify the equation: \[ 300.9326x + 303.9303 - 303.9303x = 303.001 \]Combine like terms: \[ -2.9977x + 303.9303 = 303.001 \]Isolate the variable \(x\): \[ -2.9977x = -0.9293 \]Solve for \(x\): \[ x = \frac{0.9293}{2.9977} \approx 0.31 \]So, \( x = 0.31 \), then the relative abundance of Isotope 2 is \( 1 - 0.31 = 0.69 \). The relative abundances are 31% for Isotope 1 and 69% for Isotope 2.

- Write the isotopic notations

The atomic number of wyzzlebium \(Z=120\). For the two isotopes:1. Isotope 1: \({}^{301}_{120}Wz\)2. Isotope 2: \({}^{304}_{120}Wz\)

- Calculate the number of neutrons

To find the number of neutrons in Isotope 2 (more abundant): Number of neutrons = Mass number - Atomic number \[ 304 - 120 = 184 \] There are 184 neutrons in one atom of the more abundant isotope.

Key concepts

average atomic mass

The average atomic mass is a weighted average of the masses of all isotopes of an element, reflecting their relative abundance. In simple terms, it tells us the average mass of an atom of the element, taking into account the different isotopes and how common they are. For wyzzlebium \(W z\), this value is given as \(303.001 \text{ u}\). To find this average, we consider the masses of its two isotopes, \(300.9326 \text{ u}\) and \(303.9303 \text{ u}\), along with their relative abundances. We set up an equation to solve for the relative abundances, ensuring that the sum of these abundances equals 1.

relative abundance

Relative abundance shows how common each isotope of an element is in a natural sample. It is expressed as a percentage or a fraction. For example, in wyzzlebium, we solve the equation for the average atomic mass to find the relative abundances of its isotopes. Calling the relative abundance of Isotope 1 as \(x\), and hence the abundance of Isotope 2 as \(1 - x\):
\[ 300.9326x + 303.9303(1 - x) = 303.001 \]
By solving this equation, we found that Isotope 1 has a relative abundance of \(31\text{%}\) or 0.31, and Isotope 2 has a relative abundance of \(69\text{%}\) or 0.69. This means that about 31% of wyzzlebium atoms have the mass of Isotope1, and 69% have the mass of Isotope 2.

isotopic notation

Isotopic notation is a shorthand used to denote the particular form (isotope) of an element. It includes the element’s symbol (e.g., \(W z\)), the atomic number (Z), and the mass number (A), written as \[ {}^{A}_{Z}\text{Element} \]. For wyzzlebium, with atomic number 120, and its two isotopes, we write:
\[{}^{301}_{120}Wz \] for the first isotope with a mass of \(300.9326 \text{ u}\)
\[{}^{304}_{120}Wz \] for the second isotope with a mass of \(303.9303 \text{ u}\).

neutron count

The neutron count in an atom is found by subtracting the atomic number (Z) from the mass number (A). Neutrons contribute to the mass of the atom but not to its charge. For the more abundant isotope of wyzzlebium, \[ {}^{304}_{120}\text{W z} \]:
\[ \text{Number of neutrons} = A - Z = 304 - 120 = 184 \]
Therefore, the more abundant isotope has 184 neutrons in its nucleus.

atomic number

The atomic number (Z) of an element is the number of protons in the nucleus of its atoms. It defines the element and its position in the periodic table. For wyzzlebium \(W z\), the atomic number is given as 120. This means each atom of wyzzlebium contains 120 protons. The number of protons determines the element’s chemical properties and symbolizes its identity. The atomic number remains the same across all isotopes of an element, distinguishing it from others in the periodic table.