Question

Isotope Abundance and Atomic Mass Thallium has two stable isotopes, \(^{203} \mathrm{Tl}\) and \(^{205} \mathrm{T}\). Knowing that the atomic weight of thallium is \(204.4,\) which isotope is the more abundant of the two?

Short answer

The isotope ^{205} Tl is more abundant.

Step-by-step solution

Understand Atomic Mass and Isotope Contribution

The atomic mass of an element is the weighted average of the masses of its isotopes, taking into account their abundance. Here, the isotopes are ^{203} Tl and ^{205} Tl, and we need to determine which one has a greater contribution to the average atomic mass of 204.4.

Compare Given Atomic Mass to Isotope Masses

Observe that the given atomic weight of thallium is 204.4. Compare this with the masses of the isotopes: 203 (for ^{203} Tl) and 205 (for ^{205} Tl). Since 204.4 is closer to 205, this suggests that ^{205} Tl contributes more to the atomic mass.

Conclusion on Isotope Abundance

Given the atomic weight of thallium 204.4 is closer to the mass of ^{205} Tl compared to ^{203} Tl, ^{205} Tl must be the more abundant isotope among the two. This is because its higher abundance pulls the average atomic mass closer to its own mass of 205.

Key concepts

Atomic Mass

The concept of atomic mass is fundamental when discussing elements and their isotopes. Atomic mass, often referred to as atomic weight, is essentially the average mass of all the isotopes of an element, factoring in their respective abundance.
For thallium, the atomic mass provided is 204.4. This number is not arbitrary but a calculated value representing the average weight of its isotopes.
When you think of atomic mass, picture it as a number that lies somewhere between the isotopic masses due to this averaging.

• It makes use of actual isotope masses and their natural abundance.
• Provides a comprehensive measure of an element's average weight.

Understanding atomic mass helps in comparing isotopic abundances, as seen in the case of thallium with isotopes of mass 203 and 205.

Weighted Average

A weighted average is crucial to finding out which isotope contributes more to the atomic mass. Unlike regular averages, the weighted average accounts for how much of each component, or isotope in this case, there is.
To determine the atomic mass, you multiply the mass of each isotope by its relative abundance (expressed as a fraction of 1), then sum the results. This way, more abundant isotopes sway the average closer to their own mass.

• Each isotope's mass is multiplied by its abundance.
• The sums of these products give you the atomic mass.

For thallium, the weighted average results show 204.4 due to the greater influence of the ^{205} Tl isotope, implying its higher abundance.

Isotope Contribution

Isotope contribution is all about understanding how each isotope affects the overall atomic mass of an element. It's a measure of influence based on abundance and isotopic mass. Since the atomic mass is a weighted average, the isotope with the greater abundance will have a more pronounced contribution to the atomic mass than less abundant isotopes.
In practical terms, the isotope that pulls the atomic mass closer to its own mass is the more abundant one.

• Greater isotope mass close to average implies higher abundance.
• Each isotope contributes based on its natural occurrence.

In the case of thallium, since the atomic mass 204.4 is closer to 205, we infer that ^{205} Tl has the higher abundance and thereby contributes more to the atomic mass than ^{203} Tl.