To determine the Ksp value of Hg2 I2, a chemist obtained a solid sample of Hg2 I2 in which some

To determine the Ksp value of Hg2 I2, a chemist obtained a...

To determine the $K_{\mathrm{sp}}$ value of $\mathrm{Hg}_{2} \mathrm{I}_{2},$ a chemist obtained a solid sample of $\mathrm{Hg}_{2} \mathrm{I}_{2}$ in which some of the iodine is present as radioactive $^{131}$ I. The count rate of the $\mathrm{Hg}_{2} \mathrm{I}_{2}$ sample is $5.0 \times 10^{11}$ counts per minute per mole of I. An excess amount of $\mathrm{Hg}_{2} \mathrm{I}_{2}(s)$ is placed into some water, and the solid is allowed to come to equilibrium with its respective ions. A 150.0 -mL sample of the saturated solution is withdrawn and the radioactivity measured at 33 counts per minute. From this information, calculate the $K_{\mathrm{sp}}$ value for $\mathrm{Hg}_{2} \mathrm{I}_{2}$ $$\mathrm{Hg}_{22}(s) \rightleftharpoons \mathrm{Hg}_{2}^{2+}(a q)+2 \mathrm{I}^{-}(a q) \quad K_{\mathrm{sp}}=\left[\mathrm{Hg}_{2}^{2+}\right]\left[\mathrm{I}^{-}\right]^{2}$$

Key Concepts

Conversion of Radioactivity Counts to Concentration

The process of converting radioactive count rates to ion concentration involves calibrating the measured counts against a known standard or count rate per mole of the radioactive species. This calibration allows for the determination of the amount of ion present in a sample, which can then be used in conjunction with equilibrium constants like Ksp to understand the solubility and dissociation behavior of the compound in solution.

Radioactive Tracer Techniques

Radioactive tracer techniques involve incorporating a radioisotope into a compound to track and quantify the distribution and behavior of elements or compounds in a system. By measuring the radioactivity, it becomes possible to determine the concentration of a specific ion in a solution with high sensitivity, even when the overall concentration is very low. This method is especially useful in equilibrium studies where direct measurement of ion concentrations is challenging.

Stoichiometric Relationships in Dissolution Reactions

When a compound dissolves, it dissociates into its component ions in fixed stoichiometric ratios determined by its chemical formula. Understanding these stoichiometric relationships allows for the correct formulation of the Ksp expression and connects the measured ion concentrations to the original solid phase, aiding in the calculation of solubility and other equilibrium parameters.

Saturated Solution Equilibrium

A saturated solution is one in which the maximum amount of solute has dissolved in the solvent, establishing a dynamic equilibrium between the dissolved ions and the undissolved solid. In this state, the concentrations of the ions remain constant over time as long as the temperature and pressure are unchanged. This equilibrium concept is key to applying the Ksp expression for sparingly soluble salts.

Solubility Product Constant (Ksp)

The solubility product constant, Ksp, is a measure of the extent to which a sparingly soluble compound dissolves in water. It is defined as the product of the molar concentrations of the constituent ions, each raised to the power of its stoichiometric coefficient in the dissolution reaction. Ksp is essential in predicting whether a precipitate will form under given conditions and in quantifying the solubility of ionic compounds.

Transcript

00:02 This is a ksp question that has radioactivity associated with it in order to determine ksp.

00:09 The ksp reaction is mercury 1 iodide, dissociating into the mercury 1 ion that has a 2 plus charge because it's diatomic, plus two iodides.

00:22 This equation is provided, and the ksp is going to be equal to the mercury 1 concentration, multiplied by the iodide concentration, squared.

00:33 We know that the solid before added to the water has a radioactivity of 5 .0 times 10 to the 11 counts per minute per mole of iodine.

00:46 Now it's not every single iodine that is present, that is iodine 131, that results in radioactivity, but just enough so that we can relate the activity of the ones that are radioactive to.

01:03 This is the total moles of iodine, including the ones that are not radioactive.

01:11 So if we were to add an excess amount of mercury 1 iodide, we would get a significant amount that falls to the bottom and is undissolved and a very small amount that dissolves producing two iodides for every one mercury 1.

01:29 If the volume is 150 milliliters, or if it's not and we extract a hundred and 150 milliliters, 150 mil liters saturated volume of mercury 1 iodide has a count of 33, or has a decay rate of 33 counts per minute.

01:49 What's going to be helpful as we solve this problem is to recognize that the mercury 1 concentration, based upon the stoichiometry shown up here, is going to be one -half the iodide concentration, and it is with this information the 33 counts per minute that we can determine the iodide concentration because it is only the iodide that is radioactive.

02:14 So if we take the 33 counts per minute and we multiply it by this as a conversion factor, which tells us the counts per minute per mole of radioactive iodide, and now we're just referring to the moles of radioactive iodide that have introduced themselves into the solution, we can determine the number of moles of iodide that actually introduced themselves into the solution...

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