Question

The radioactive element technetium is not found naturally on earth; it must be synthesized in the laboratory. It is a valuable element, however, because it has medical uses. For example, the element in the form of sodium pertechnetate \(\left(\mathrm{NaTcO}_{4}\right)\) is used in imaging studies of the brain, thyroid, and salivary glands and in renal blood flow studies, among other things. (a) In what group and period of the periodic table is the element found? (b) The valence electrons of technetium are found in the \(5 s\) and \(4 d\) subshells. What is a set of quantum numbers \(\left(n, \ell, \text { and } m_{\ell}\right)\) for one of the electrons of the \(5 s\) subshell? (c) Technetium emits a \(\gamma\) -ray with an energy of \(0.141 \mathrm{MeV} (1 \mathrm{MeV}=10^{6}\) electron-volts, where $$\left.1 \mathrm{eV}=1.6022 \times 10^{-19} \mathrm{J} .\right) \text { What are the wavelength }$$and frequency of a \(\gamma\) -ray photon with an energy of \(0.141 \mathrm{MeV} ?\) (d) To make \(\mathrm{NaTcO}_{4},\) the metal is dissolved in nitric acid. $$\begin{aligned} 7 \mathrm{HNO}_{3}(\mathrm{aq})+& \mathrm{Tc}(\mathrm{s}) \rightarrow \\ & \mathrm{HTcO}_{4}(\mathrm{aq})+7 \mathrm{NO}_{2}(\mathrm{g})+3 \mathrm{H}_{2} \mathrm{O}(\ell) \end{aligned}$$ and the product, HTcO \(_{4}\), is treated with \(\mathrm{NaOH}\) to make \(\mathrm{NaTcO}_{4}\) (i) Write a balanced equation for the reaction of \(\mathrm{HTcO}_{4}\) with \(\mathrm{NaOH}\) (ii) If you begin with 4.5 mg of Tc metal, what mass of \(\mathrm{NaTcO}_{4}\) can be made? What mass of \(\mathrm{NaOH}\), in grams, is required to convert all of the \(\mathrm{HTcO}_{4}\) into \(\mathrm{NaTcO}_{4} ?\) (e) If you synthesize 1.5 micromoles of \(\mathrm{NaTcO}_{4},\) what mass of compound do you have? If the compound is dissolved in \(10.0 \mathrm{mL}\) of solution, what is the concentration?

Short answer

(a) Group 7, Period 5. (b) \((5, 0, 0)\). (c) \( \lambda = 8.79 \times 10^{-12} \text{ m}\), \(f = 3.412 \times 10^{19} \text{ Hz}\). (d.i) \(\text{HTcO}_4 + \text{NaOH} \rightarrow \text{NaTcO}_4 + \text{H}_2\text{O}\). (d.ii) 7.46 mg \(\text{NaTcO}_4\); 1.82 mg \(\text{NaOH} \). (e) 0.246 mg; 0.15 M.

Step-by-step solution

(a) Determine the Group and Period

Technetium's atomic number is 43. It belongs to the d-block elements known as the transition metals. Technetium is found in Group 7 and Period 5 of the periodic table.

(b) Identify Quantum Numbers for 5s Electron

The quantum numbers for the 5s subshell can be found as follows:- Principal quantum number, \(n = 5\).- Azimuthal quantum number, \(\ell = 0\) (since s subshell corresponds to \(\ell = 0\)).- Magnetic quantum number, \(m_{\ell} = 0\) (as \(m_{\ell}\) can range from \(-\ell\) to \(+\ell\)).Thus, a possible set of quantum numbers for a 5s electron is \((5, 0, 0)\).

(c) Calculate Wavelength and Frequency of γ-ray Photon

To determine wavelength \(\lambda\) and frequency \(f\), we first convert energy from MeV to Joules:\[ E = 0.141 \text{ MeV} = 0.141 \times 10^6 \times 1.6022 \times 10^{-19} \text{ J/eV} = 2.2611 \times 10^{-14} \text{ J} \]Using \( E = hf \), where \( h = 6.626 \times 10^{-34} \text{ Js} \), solve for \( f \):\[ f = \frac{E}{h} = \frac{2.2611 \times 10^{-14}}{6.626 \times 10^{-34}} = 3.412 \times 10^{19} \text{ Hz} \]Using \( c = f\lambda \), where \( c = 3.00 \times 10^8 \text{ m/s} \), solve for \( \lambda \):\[ \lambda = \frac{c}{f} = \frac{3.00 \times 10^8}{3.412 \times 10^{19}} = 8.79 \times 10^{-12} \text{ m} \]

(d.i) Write Balanced Equation for NaTcO₄ Formation

The reaction between \( \text{HTcO}_4 \) and \( \text{NaOH} \) is given by:\[ \text{HTcO}_4 (\text{aq}) + \text{NaOH} (\text{aq}) \rightarrow \text{NaTcO}_4 (\text{aq}) + \text{H}_2\text{O} (\ell) \]

(d.ii) Calculate Mass of NaTcO₄ and NaOH Requirement

First, determine the moles of Tc given 4.5 mg:\[ \text{Moles of Tc} = \frac{4.5 \times 10^{-3} \text{ g}}{98.906 \text{ g/mol}} = 4.55 \times 10^{-5} \text{ mol} \]Assuming 1:1 molar conversion for Tc to \( \text{NaTcO}_4 \):\[ \text{Mass of } \text{NaTcO}_4 = 4.55 \times 10^{-5} \text{ mol} \times 163.92 \text{ g/mol} = 7.46 \times 10^{-3} \text{ g} \]For \( \text{NaOH} \), use 1:1 molar conversion:\[ \text{Mass of NaOH} = 4.55 \times 10^{-5} \text{ mol} \times 40.00 \text{ g/mol} = 1.82 \times 10^{-3} \text{ g} \]

(e) Calculate Mass and Concentration of NaTcO₄

To determine mass for 1.5 micromoles:\[ \text{Mass of NaTcO}_4 = 1.5 \times 10^{-6} \text{ mol} \times 163.92 \text{ g/mol} = 2.46 \times 10^{-4} \text{ g} \]Concentration when dissolved in 10 mL solution:\[ \text{Concentration} = \frac{1.5 \times 10^{-6}}{10 \times 10^{-3}} = 0.15 \text{ M} \]

Key concepts

Radioactive elements

Technetium is a fascinating element because it does not exist naturally on Earth. Instead, it is synthesized in laboratories. Like some other radioactive elements, technetium has unstable nuclei that release energy in the form of radiation. This radioactive decay is what makes technetium particularly useful in certain applications.

Its radioactive properties allow it to emit \( \gamma \) -rays, which are high-energy photons. These emissions are useful in various technological and medical applications, helping us to explore and understand different physical processes. When handling radioactive elements like technetium, safety is paramount due to their radiation.

Transition metals

Technetium is part of a group called transition metals. These elements are known for having partially filled d orbitals. They're usually capable of forming colorful compounds and possess unique chemical behaviors.

Transition metals, including technetium, are located in the d-block of the periodic table. Specifically, technetium can be found in Group 7 and Period 5.

• They often have high melting points.
• They're good conductors of electricity.
• They show a variety of oxidation states.

These properties make transition metals highly versatile and important in various industrial and chemical processes.

Quantum numbers

Quantum numbers are fundamental to understanding the behavior of electrons in atoms. For each electron in an atom, a unique set of quantum numbers is assigned, which helps define its state and position.

There are four quantum numbers:

• Principal quantum number \( n \): Indicates the energy level of the electron. For a 5s electron in technetium, \( n = 5 \).
• Azimuthal quantum number \( \ell \): Defines the shape of the orbital. For the 5s subshell, \( \ell = 0 \).
• Magnetic quantum number \( m_{ \ell }\): Indicates the orientation of the orbital in space. For \( \ell = 0 \), \( m_{ \ell } = 0 \).
• Spin quantum number: Defines the direction of the electron's spin, either +1/2 or -1/2.

For the 5s electron in technetium, a possible set of quantum numbers would be (5, 0, 0, +1/2) or (5, 0, 0, -1/2).

Chemical synthesis

Chemical synthesis refers to the process of creating chemical compounds from simpler materials. Technetium cannot be found naturally, so it must be synthesized through chemical reactions in a lab.

In the synthesis of sodium pertechnetate (\( \text{NaTcO}_4 \)), technetium metal reacts with nitric acid, as shown in the equation:

\[ 7 \text{HNO}_3(\text{aq}) + \text{Tc}(\text{s}) \rightarrow \text{HTcO}_4(\text{aq}) + 7 \text{NO}_2(\text{g}) + 3 \text{H}_2 \text{O}(\ell) \]
This sequence of reactions is carefully controlled to produce the desired compound. Chemical synthesis is vital in creating specific compounds for applications in medicine, industry, and research.

Medical imaging applications

Technetium's remarkable properties make it invaluable in medical imaging. Its radioactive nature allows it to emit \( \gamma \) -rays, which are crucial in diagnostic imaging.

In healthcare, technetium-99m, a specific isotope, is used in procedures like:

• Brain imaging
• Thyroid scans
• Renal blood flow studies

These imaging techniques enable doctors to see inside the body without invasive procedures. The emitted radiation is captured using special cameras, providing detailed images that help diagnose and monitor various health conditions.