Problem 158 - Chemistry and Chemical Reactivity

Question

Mass spectrometric analysis showed that there are four isotopes of an unknown element having the following masses and abundances: $$\begin{array}{|c|c|c|c|}\hline \text { Isotope } & \begin{array}{c}\text { Mass } \\\\\text { Number }\end{array} & \begin{array}{c}\text { Isotope } \\\\\text { Mass }\end{array} & \begin{array}{c}\text { Abundance } \\\\(\%)\end{array} \\\\\hline 1 & 136 & 135.9090 & 0.193 \\\\\hline 2 & 138 & 137.9057 & 0.250 \\\\\hline 3 & 140 & 139.9053 & 88.48 \\\\\hline 4 & 142 & 141.9090 & 11.07 \\\\\hline\end{array}$$ Three elements in the periodic table that have atomic weights near these values are lanthanum (La), atomic number \(57,\) atomic weight 138.9055 cerium (Ce), atomic number \(58,\) atomic weight \(140.115 ;\) and praseodymium \((\mathrm{Pr}),\) atomic number \(59,\) atomic weight \(140.9076 .\) Using the data above, calculate the atomic weight, and identify the element if possible.

Step-by-Step Solution

Answer

The element is likely Cerium (Ce).

Step 1: Calculate the Fractional Abundances

First, convert the percent abundances into decimal fractions by dividing by 100.\[\text{Fractional Abundance}_1 = \frac{0.193}{100} = 0.00193\] \[\text{Fractional Abundance}_2 = \frac{0.250}{100} = 0.00250\] \[\text{Fractional Abundance}_3 = \frac{88.48}{100} = 0.8848\] \[\text{Fractional Abundance}_4 = \frac{11.07}{100} = 0.1107\]

Step 2: Calculate Contribution to Atomic Weight

Multiply each isotope's mass by its fractional abundance to find its contribution to the atomic weight:\[\text{Contribution}_1 = 135.9090 \times 0.00193 = 0.2623\] \[\text{Contribution}_2 = 137.9057 \times 0.00250 = 0.3448\] \[\text{Contribution}_3 = 139.9053 \times 0.8848 = 123.7303\] \[\text{Contribution}_4 = 141.9090 \times 0.1107 = 15.7092\]

Step 3: Summing Contributions for Total Atomic Weight

Add all contributions together to find the total atomic weight:\[\text{Atomic Weight} = 0.2623 + 0.3448 + 123.7303 + 15.7092 = 140.0466\]

Step 4: Identify the Element

Compare the calculated atomic weight (140.0466) with the atomic weights of potential elements. The calculated weight is closest to that of cerium (Ce), which has an atomic weight of 140.115.

Key Concepts

Mass Spectrometry

Mass spectrometry is an analytical technique that helps scientists determine the masses of isotopes within a sample. Think of it as a tool that weighs atoms. It works by ionizing chemical compounds to generate charged molecules or molecule fragments and measuring their mass-to-charge ratios. Here's a quick look at the main steps of mass spectrometry:

• The sample is vaporized and then ionized to form charged particles.
• These ions are accelerated by an electric field, making them move.
• The ions pass through a magnetic field, which deflects them based on their mass-to-charge ratio.
• The deflected ions hit a detector, providing data about the isotopes present.

With mass spectrometry, scientists can identify the specific isotopes an element contains, as well as their relative amounts in a sample. This forms the foundation for understanding the isotopic composition of elements.

Atomic Weight Calculation

The atomic weight of an element is a weighted average of the masses of its isotopes. This value reflects the masses and relative abundances of the naturally occurring isotopes of that element. The calculation involves several steps:

• Convert the percentage abundance of each isotope to fractional abundance by dividing by 100.
• Multiply the mass of each isotope by its fractional abundance to find its contribution to the atomic weight.
• Sum all these contributions to get the total atomic weight.

Let's take a closer look at how this applies practically. Suppose you have an element with various isotopes, each contributing differently to the atomic weight. By calculating each isotope's contribution and combining them, you achieve an average atomic weight, essential for identifying the element or understanding its characteristics.

Fractional Abundance

Fractional abundance is a way to express the percentage of each isotope in decimal form. It provides a clearer picture when you calculate atomic weights. To find fractional abundance, convert each isotope's abundance percentage by dividing it by 100. For example:

• A 0.193% abundance becomes 0.00193 as a fractional abundance.
• This small change makes calculations easier and more accurate.

Fractional abundance allows scientists to weigh the significance of each isotope's contribution to the element’s overall atomic weight. By transforming percentage to a fraction, you can carry out precise mathematical operations like multiplication with isotopic masses. This is a crucial step in determining the atomic weight.

Isotopes Identification

Identifying isotopes involves recognizing the different forms of atoms that comprise an element, each defined by a different number of neutrons. Isotopes have nearly identical chemical properties but differ in mass. Here's how you identify isotopes:

• Lighter or heavier forms of an element are noted by their mass number—sum of protons and neutrons.
• Mass spectrometry is used to differentiate and identify each isotope of an element.
• This technique measures the mass and relative abundance of isotopes, providing a spectrum showing the mass-to-charge ratios.

Identifying isotopes is vital for understanding elements' behaviors and applications in fields like chemistry and physics, where knowing exact isotopic compositions influences reactions and other properties. In the exercise, the isotopes of a mysterious element were revealed, and comparing their atomic weight to known elements helped pinpoint that the unknown element is cerium.