Question

Antimony (Sb), which has an atomic weight of \(121.75\) amu, has two naturally occurring isotopes: Sb-121 and Sb-123. If a sample of antimony is \(42.70 \% \mathrm{Sb}-123\), which has a mass of \(122.90 \mathrm{amu}\), what is the mass of Sb-121?

Short answer

The mass of Sb-121 is approximately 120.91 amu.

Step-by-step solution

- Identify the known values

Write down the known values:- Atomic weight of Antimony (Sb): 121.75 amu- Mass of Sb-123 isotope: 122.90 amu- Percentage of Sb-123 in the sample: 42.70%- Percentage of Sb-121 in the sample: 100% - 42.70% = 57.30%

- Set up the equation

Use the weighted average formula for atomic weight:Atomic weight of Sb = (Fraction of Sb-121 × Mass of Sb-121) + (Fraction of Sb-123 × Mass of Sb-123)Substitute the known values: 121.75 = (0.5730 × Mass of Sb-121) + (0.4270 × 122.90)

- Isolate the variable

Rearrange the equation to solve for the mass of Sb-121:121.75 = 0.5730× Mass of Sb-121 + 52.4563Subtract 52.4563 from both sides:69.2937 = 0.5730 × Mass of Sb-121Divide both sides by 0.5730:Mass of Sb-121 = 69.2937 / 0.5730

- Perform the calculation

Calculate the mass of Sb-121:Mass of Sb-121 ≈ 120.91 amu

Key concepts

Isotopes

Isotopes are different forms of the same element that contain the same number of protons but a different number of neutrons in their nuclei. This difference in neutron count means isotopes of an element have different masses. For example, antimony (Sb) has two main isotopes—Sb-121 and Sb-123. While both isotopes have the same chemical properties, their different masses allow scientists to distinguish between them using precise measurements.

Weighted Average

The atomic weight of an element is a weighted average of the masses of its isotopes. This means it is calculated by considering both the mass and the relative abundance of each isotope. The weighted average gives a value that accounts for all isotopic variations naturally occurring in a sample. For antimony, this calculation involves combining the masses of Sb-121 and Sb-123, weighted by their respective percentages in a typical sample.

Mass Calculation

To find the mass of a particular isotope, we can use the concept of the weighted average along with some algebra. For antimony, if we know the atomic weight (121.75 amu), the mass of Sb-123 (122.90 amu), and the percentage of Sb-123 in the sample (42.70%), we can set up an equation to solve for the mass of Sb-121. The equation combines these values to isolate the unknown mass value.

Sb-121

Sb-121 is one of the naturally occurring isotopes of antimony. In a sample where Sb-123 makes up 42.70%, Sb-121 must account for the remaining 57.30%. Using the weighted average formula, and substituting known values, we solve for the mass of Sb-121. In the given exercise, the mass calculation leads to a result of approximately 120.91 amu for Sb-121.

Sb-123

Sb-123 is the second naturally occurring isotope of antimony, with a mass of 122.90 amu. It makes up 42.70% of the sample. Combining this information with the atomic weight of antimony and the percentage of Sb-121, we can find the weighted contribution of Sb-123 to the overall atomic weight. These values are crucial in isolating and calculating the mass of Sb-121 using algebraic manipulation.