Question

A complex ion of chromium(III) with oxalate ion was prepared from \({ }^{51} \mathrm{Cr}\) -labeled \(\mathrm{K}_{2} \mathrm{Cr}_{2} \mathrm{O}_{7},\) having a specific activity of \(843 \mathrm{cpm} / \mathrm{g}\) (counts per minute per gram), and \({ }^{14} \mathrm{C}\) -labeled oxalic acid, \(\mathrm{H}_{2} \mathrm{C}_{2} \mathrm{O}_{4}\), having an specific activity of \(345 \mathrm{cpm} / \mathrm{g}\). Chromium- 51 decays by electron capture with the emission of gamma radiation, whereas \({ }^{14} \mathrm{C}\) is a pure beta emitter. Because of the characteristics of the beta and gamma detectors, each of these isotopes may be counted independently. A sample of the complex ion was observed to give a gamma count of 165 cpm and a beta count of 83 cpm. From these data, determine the number of oxalate ions bound to each \(\mathrm{Cr}(\mathrm{III})\) in the complex ion. (Hint: \(\mathrm{For}\) the starting materials calculate the cpm per mole of \(\mathrm{Cr}\) and oxalate, respectively.)

Short answer

The number of oxalate ions bound to each Cr(III) in the complex ion based on the given counts.

Step-by-step solution

Determine the Cr activity per mole

First, calculate the activity of the Chromium-

Calculate the counts per mole of Chromium in the sample

To find the number

Determine the oxalate activity per mole

Next, calculate the activity of the oxalate ion (C2O4) in the sample...

Calculate the counts per mole of Oxalate in the sample

Calculate the counts per mole of oxalate in the sample using the given specific activity and the observed beta count...

Find the ratio of counts per mole of Oxalate to Chromium

Find the ratio of the number of oxalates per chromium atom by dividing the counts from Step 4 by the counts from Step 2...

Determine the number of oxalate ions per Cr(III)

Since we know the ratio of counts per mole corresponds to the ratio of number of ions, we can determine the number of oxalate ions attached to each Cr(III) based on the ratio calculated in Step 5.

Key concepts

Radioactive Isotopes in Chemistry

Radioactive isotopes, also known as radionuclides, are atoms with an unstable nucleus that release energy to reach a more stable state. This release of energy is what we refer to as radioactivity. In chemistry, these isotopes serve as powerful tools in various fields including medical diagnostics, treatment, and research.

For instance, Chromium-51 (^{51}Cr) and Carbon-14 (^{14}C), mentioned in our textbook exercise, are two such isotopes. Chromium-51 decays by electron capture, which involves the capture of an electron by the nucleus and results in the emission of gamma radiation—an energetic form of electromagnetic radiation. In contrast, Carbon-14 undergoes beta decay, where a beta particle (an electron or a positron) is emitted.

The properties of these isotopes allow scientists to trace the movement and concentration of specific elements or compounds in both living and non-living systems. By measuring the radiation emitted (in counts per minute or cpm), it's possible to gain insights into complex chemical and biological processes, like the formation of a chromium oxalate complex in our exercise.

Chromium Oxalate Complex

A complex ion involves a metal ion at the center surrounded by molecules or anions, known as ligands, which can donate electrons to the metal ion. These formations are a focal point in coordination chemistry.

In the example of the chromium oxalate complex from our textbook problem, the chromium(III) ion forms a complex with the oxalate ion (C_2O_4^2-). Each ligand bonds to the chromium through a process called chelation, which involves the formation of multiple bonds with the central metal ion. The study of these complexes is not only important for understanding fundamental chemistry but also for practical applications such as in industrial catalysis or environmental remediation.

The determination of the number of oxalate ions bound to each chromium ion in such a complex typically involves experimental measurements where the properties of the complex can be deduced from the behavior of its radioactive isotopes.

Mole Concept in Chemistry

The mole is a fundamental concept in chemistry that refers to a specific quantity of particles, be it atoms, molecules, ions, or electrons. One mole is defined as the number of atoms in exactly 12 grams of pure carbon-12, which is approximately 6.022 x 10^23 particles— a value known as Avogadro’s number.

Understanding the mole concept is vital when dealing with chemical reactions. It allows chemists to measure substances in a consistent way, providing a bridge between the atomic scale and the macroscopic world. When we calculate specific activity as in the exercise (cpm per gram), we extend this to measure the activity of radioactive isotopes in a given quantity; however, the true goal is often to determine activity per mole, which then allows us to explore the stoichiometry of chemical reactions and complex formation.

In our textbook exercise, by calculating the specific activity per mole for both Chromium-51 and Carbon-14, we can deduce important stoichiometric relationships in the formation of the chromium oxalate complex.