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: $$\begin{array}{cccc} \text { Isotope } & \text { Mass Number } & \text { Isotope Mass } & \text { Abundance (\%) } \\ \hline 1 & 136 & 135.9090 & 0.193 \\ 2 & 138 & 137.9057 & 0.250 \\ 3 & 140 & 139.9053 & 88.48 \\ 4 & 142 & 141.9090 & 11.07 \end{array}$$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 praeseodymium (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 Cerium (Ce) with an atomic weight of 140.1346.

Step 1: Multiply Isotope Mass by Abundance

For each isotope, multiply the isotopic mass by its abundance (in decimal form) to get the contributed weight of that isotope to the atomic weight.- Isotope 1: \(135.9090 \times 0.00193 = 0.2623\)- Isotope 2: \(137.9057 \times 0.00250 = 0.3448\)- Isotope 3: \(139.9053 \times 0.8848 = 123.8143\)- Isotope 4: \(141.9090 \times 0.1107 = 15.7132\)

Step 2: Calculate the Total Atomic Weight

Add up the contributions from each isotope calculated in Step 1 to find the atomic weight of the element.\[0.2623 + 0.3448 + 123.8143 + 15.7132 = 140.1346\]

Step 3: Identify the Element

Compare the calculated atomic weight with the atomic weights of the provided elements (La, Ce, Pr) to identify the element. - Lanthanum (La): 138.9055 - Cerium (Ce): 140.115 - Praeseodymium (Pr): 140.9076 The calculated atomic weight, 140.1346, is closest to that of Cerium (Ce).

Key Concepts

Mass Spectrometry

Mass spectrometry is a powerful technique used in chemistry to analyze the composition of elements and their isotopes. It works by ionizing chemical compounds to generate charged molecules or molecule fragments and then measuring their mass-to-charge ratios. This process allows scientists to identify different isotopes of an element by their distinct mass.
When you conduct mass spectrometric analyses, the machinery will generate a mass spectrum — a graph depicting the mass-to-charge ratio on the x-axis and relative abundance on the y-axis. This spectrum helps in determining the number of isotopes, as well as their respective masses and abundances, which is crucial for further calculations such as atomic weight.
In the exercise, mass spectrometry provided the data for four isotopes with specific masses and their percentage abundances, acting as the foundation for calculating the atomic weight.

Atomic Weight Calculation

The atomic weight of an element is an average of the isotopic masses of its various isotopes, weighted by their relative natural abundances. The formula to calculate this is straightforward but essential: the atomic weight is the sum of the products of each isotope's mass and its relative abundance converted into decimal form.
To perform this calculation:

• Convert each isotope's percentage abundance into decimal form.
• Multiply each isotope's mass by this decimal abundance to get its contribution.
• Add up these contributions to find the total atomic weight.

For instance, in our exercise, isotope 3 with a large abundance of 88.48% contributes significantly more to the atomic weight than the others. The final atomic weight is 140.1346, a precise value pointing towards the element type when compared to known atomic weights.

Isotopes

Isotopes are variants of a particular chemical element that have the same number of protons but different numbers of neutrons. This variation does not affect the chemical behavior of the substance significantly but does change its atomic mass and other properties.
In our exercise, isotopes of the unknown element were discovered through mass spectrometry. They showed that even though these atoms belonged to the same element, each isotope had slight differences in mass due to changes in neutron number. These differences are critical when calculating the weighted atomic weight, as a more naturally abundant isotope will skew the average mass towards its value.
Understanding isotopes is crucial for accurate element identification and supports numerous scientific fields, including geology, medicine, and archaeology, where isotopic analysis can reveal much about historical and environmental conditions.

Element Identification

Once the atomic weight is calculated, identifying the element involves comparing this weight with known atomic weights of potential candidates. Elements near the calculated weight are considered. This exercise involves matching the computed atomic weight of 140.1346 with the atomic weights of lanthanum (La), cerium (Ce), and praeseodymium (Pr).
The process is similar to connecting dots. The closer the matched atomic weight to a known element's weight, the more likely you've correctly identified the element. Here, the computed weight was closest to that of cerium (Ce), a process of elimination vital in scientific inquiries and educational practices.
This step in elemental analysis is indispensable for anyone working with chemical substances, as knowing the exact element can lead to understanding its properties and potential reactivity.