Question

The amount of oxygen dissolved in a sample of water can be determined by using thallium metal containing a small amount of the isotope Tl-204. When excess thallium is added to oxygen-containing water, the following reaction occurs. $$ 2 \mathrm{Tl}(s)+\frac{1}{2} \mathrm{O}_{2}(g)+\mathrm{H}_{2} \mathrm{O} \longrightarrow 2 \mathrm{Tl}^{+}(a q)+2 \mathrm{OH}^{-}(a q) $$ After reaction, the activity of a 25.0 -mL water sample is 745 counts per minute (cpm), caused by the presence of \(\mathrm{Tl}^{+}-204\) ions. The activity of \(\mathrm{Tl}-204\) is \(5.53 \times 10^{5} \mathrm{cpm}\) per gram of thallium metal. Assuming that \(\mathrm{O}_{2}\) is the limiting reactant in the above equation, calculate its concentration in moles per liter.

Short answer

Question: Determine the concentration of O₂ in a 25.0 mL water sample, given that the activity of the water sample after adding excess thallium is 745 counts per minute (cpm). The activity of Tl-204 is \(5.53 \times 10^5\) cpm per gram of thallium metal, and the reaction stoichiometry is 2 moles of Tl to 0.5 moles of O₂.

Step-by-step solution

Find the moles of Tl-204 in the reaction

The activity of the water sample is 745 cpm, and we are given the activity of Tl-204 is \(5.53 \times 10^5\) cpm per gram of thallium metal. We can use this information to find the amount of Tl-204 in the reaction. \(\frac{745 \ \text{cpm} }{5.53\times 10^5\ \text{cpm/g}} = m\) (mass of Tl-204 in g)

Convert the mass of Tl-204 to moles

Calculate the number of moles of Tl-204 using its molar mass (204.38 g/mol): \(moles \text{ of Tl-204} = \frac{m}{204.38\ \text{g/mol}}\)

Determine moles of O₂ reacted

Since the stoichiometry of the reaction is 2 moles of Tl to 0.5 moles of O₂, we can use the moles of Tl-204 we found to determine the moles of O₂ reacted. \(moles\ \text{of O}_{2} = \frac{1}{2} \times moles\ \text{of Tl-204}\)

Determine O₂ concentration

Finally, we can find the concentration of O₂ in the water sample by dividing the moles of O₂ by the volume of the sample. \(\text{O}_{2} \ \text{Concentration} = \frac{moles\ \text{of O}_{2} }{0.025\ \text{L}}\) Calculate the concentration keeping the appropriate number of significant figures. The result will be in moles of O₂ per liter.