2026 Free Response

Consider the $\text{K}^+$ ion in KCl.

Write the complete ground-state electron configuration for the potassium ion, $\text{K}^+$.

Which has the larger radius, the $\text{K}^+$ ion or the K atom? Explain your reasoning using principles of atomic structure.

The following information applies to parts B, C, D, and E.

A student performs an experiment to determine the enthalpy of solution, $\Delta H_{soln}$, of KCl. The student places water in a calorimeter, measures the initial temperature, adds $\text{KCl}(s)$, and stirs until the KCl dissolves completely. The temperature is continuously monitored during the experiment. Data from the experiment are given in the following table.

Describe how the student can use the temperature readings during the experiment to determine when the dissolution is complete.

Use the data in the table to do the following.

Calculate the magnitude of the thermal energy, $q$, in joules, transferred during the dissolution. Show the work that leads to your answer. (Assume that the solution has a specific heat capacity of $3.95\ \text{J}/(\text{g}\cdot°\text{C})$.)

Calculate the value of the molar enthalpy of solution, $\Delta H_{soln}$, in kJ/mol, for KCl given that $0.0912\ \text{mol}$ of KCl dissolved. Show the work that leads to your answer. Include the sign with your answer.

The calorimeter is not perfectly insulated. Would this cause the magnitude of $\Delta H_{soln}$ calculated in part C (ii) to be greater than, less than, or equal to the accepted value? Justify your answer.

The student performs a second experiment using the same mass of water but $6.80\ \text{g}$ of $\text{RbCl}(s)$ instead of $6.80\ \text{g}$ of $\text{KCl}(s)$. The magnitude of the molar enthalpy of solution, $\Delta H_{soln}$, of RbCl is approximately equal to that of KCl.

Is the magnitude of $\Delta T$ in the RbCl experiment greater than, less than, or equal to that in the KCl experiment? Justify your answer. (Assume that the specific heat capacities of the solutions are equal.)

RbCl has a $K_{sp}$ of 57 at $20°\text{C}$.

Write the net ionic equation for the dissolution of RbCl in pure water.

Calculate the molar solubility of RbCl in a saturated solution at $20°\text{C}$. Show the work that leads to your answer.

A second saturated solution is prepared by dissolving $\text{RbCl}(s)$ in $1.0\ M\ \text{KCl}(aq)$ at $20°\text{C}$ instead of in pure water. Will the molar solubility of RbCl in $1.0\ M\ \text{KCl}(aq)$ be greater than, less than, or equal to the molar solubility calculated in part F (ii)? Justify your answer.

The bonds between chromium and oxygen in $\text{CrO}_4^{2-}$ can be represented as covalent bonds. One possible Lewis diagram for the $\text{CrO}_4^{2-}$ ion is shown in Figure 1.

[Figure 1: Lewis diagram of $\text{CrO}_4^{2-}$ — a central Cr atom single-bonded to three O atoms (each with three lone pairs) and double-bonded to one O atom (with two lone pairs), the whole structure enclosed in brackets with a $2-$ charge.]

Based on VSEPR theory, predict the molecular geometry of the $\text{CrO}_4^{2-}$ ion.

On Figure 2, draw a complete Lewis diagram for a resonance structure of $\text{CrO}_4^{2-}$ that results in a formal charge of zero on two of the O atoms and a formal charge of $-1$ on the other two O atoms. Your diagram should contain the same number of valence electrons as in Figure 1.

[Figure 2: A blank template showing Cr in the center bonded to four O atoms arranged above, left, right, and below, enclosed in brackets with a $2-$ charge, for the student to complete.]

When reacted with a strong acid like $\text{HNO}_3(aq)$, $\text{CrO}_4^{2-}(aq)$ can be converted to $\text{Cr}_2\text{O}_7^{2-}(aq)$ and $\text{H}_2\text{O}(l)$ in a reversible reaction.

Write a balanced net ionic equation for the reaction between $\text{CrO}_4^{2-}(aq)$ and a strong acid.

Is the reaction a redox reaction? Justify your answer based on the oxidation number of Cr.

The following information applies to parts C and D.

In acidic solutions, $\text{Cr}_2\text{O}_7^{2-}(aq)$ reacts to form $\text{CrO}_3(aq)$ according to Equation 1.

$\text{CrO}_3(aq)$ can be electrolyzed to plate objects with a thin layer of chromium metal according to Equation 2.

Is the reaction represented by Equation 2 thermodynamically favorable or unfavorable under standard conditions? Justify your answer with a calculation of $\Delta G°$. Show the work that leads to your answer.

Calculate the number of grams of $\text{Cr}(s)$ that could be plated onto the surface of a steel rod when $15.0\ \text{A}$ of current is applied for $3250$ seconds. Show the work that leads to your answer.

In a separate experiment, an iron wire is placed directly in a solution of $0.50\ M\ \text{Cr}_2\text{O}_7^{2-}(aq)$ with $\text{pH} = 1.99$. The concentration of $\text{Cr}_2\text{O}_7^{2-}(aq)$ remaining is recorded over time, and the results are presented in Figure 3.

[Figure 3: Graph of $[\text{Cr}_2\text{O}_7^{2-}]$ (M) vs. Time (minutes); y-axis 0.00 to 0.50 in increments of 0.05, x-axis 0 to 48 in increments of 4; curve is concave up, decreasing from approximately $(0, 0.50)$ through $(4, 0.44)$, $(8, 0.38)$, $(12, 0.33)$, $(16, 0.29)$, $(20, 0.26)$, $(24, 0.23)$, $(28, 0.21)$, $(32, 0.18)$, $(36, 0.16)$, $(40, 0.14)$, $(44, 0.12)$, to approximately $(48, 0.10)$.]

Explain how the data in Figure 3 support the conclusion that the reaction is first order with respect to $\text{Cr}_2\text{O}_7^{2-}$.

The following information applies to parts F and G.

The student creates a plot of $\ln[\text{Cr}_2\text{O}_7^{2-}]$ versus time for a trial with an initial concentration of $0.50\ M\ \text{Cr}_2\text{O}_7^{2-}(aq)$, as shown in Figure 4.

[Figure 4: Graph of $\ln[\text{Cr}_2\text{O}_7^{2-}]$ vs. Time (minutes); y-axis $-2.40$ to $0.00$ in increments of $0.20$, x-axis 0 to 18 in increments of 3; dashed line of best fit through points starting near $(0, -0.65)$ decreasing roughly linearly to approximately $(18, -1.35)$.]

Calculate the value of the rate constant, $k$, for the reaction. Report your answer in units of $\text{min}^{-1}$. Show the work that leads to your answer.

The student then runs a second trial of the experiment under identical conditions but uses an initial concentration of $0.25\ M\ \text{Cr}_2\text{O}_7^{2-}(aq)$ instead of $0.50\ M\ \text{Cr}_2\text{O}_7^{2-}(aq)$. On Figure 4, carefully draw the plot of $\ln[\text{Cr}_2\text{O}_7^{2-}]$ versus time that would be expected for the second experiment. The point at time zero has already been plotted.

Identify a conjugate acid-base pair in the equation. Be sure to clearly label which is the acid and which is the base.

A solution of $\text{HNO}_2(aq)$ with an initial concentration of $0.125\ M$ has a pH of 2.09.

Calculate the value of $[\text{H}_3\text{O}^+]$ in the solution. Show the work that leads to your answer.

Calculate $[\text{HNO}_2]$ at equilibrium. Show the work that leads to your answer.

In an experiment, the solution of $0.125\ M\ \text{HNO}_2(aq)$ is heated to 333 K. The new equilibrium concentrations are determined to be $[\text{HNO}_2] = 0.114\ M$, $[\text{NO}_2^-] = 0.0109\ M$, and $[\text{H}_3\text{O}^+] = 0.0109\ M$.

Calculate the value of $K_a$ for $\text{HNO}_2$ at 333 K. Show the work that leads to your answer.

Is the reaction represented by Equation 1 endothermic or exothermic? Justify your answer by comparing $K_a$ values at 298 K and 333 K.

The following information applies to parts D, E, and F.

In a second experiment, a student is given a bottle containing $\text{HNO}_2$ with an unknown molarity. To determine the concentration of the $\text{HNO}_2(aq)$ in the bottle, the student titrates $35.0\ \text{mL}$ of the $\text{HNO}_2(aq)$ with $0.16\ M\ \text{NaOH}(aq)$ at 298 K, which reacts according to Equation 2.

The results of the titration are summarized in Figure 1.

[Figure 1: Titration curve, pH vs. Volume of 0.16 M NaOH Added (mL); y-axis pH 0 to 14 in increments of 2, x-axis 0 to 80 mL in increments of 10; curve starts near $(0, 2)$, rises gradually to about $(40, 4)$, then rises steeply (equivalence point) between about 45-55 mL up to about pH 12, then levels off gradually to approximately $(80, 12.7)$.]

Using Figure 1, identify the pH at the equivalence point.

Using Figure 1, calculate the molarity $\text{HNO}_2(aq)$ of in the bottle. Show the work that leads to your answer.

Draw an X on Figure 1 to represent a point in the titration where $[\text{HNO}_2] > [\text{NO}_2^-]$ in the reaction mixture.

Equilibrium constants at 298 K are shown in Table 1 for the acid ionization reaction represented by Equation 1, $K_a$; the autoionization of water, $K_w$; and the neutralization reaction represented by Equation 2, $K_2$.

Calculate the value of $K_2$ for Equation 2 at 298 K. Show the work that leads to your answer.

The student decides to repeat the experiment, this time using an indicator, and has access to the indicators in Table 2.

The student conducts a second titration by adding two drops of methyl orange indicator to $35.0\ \text{mL}$ of the $\text{HNO}_2(aq)$ solution and titrating the solution with $0.16\ M\ \text{NaOH}(aq)$ until a color change from red to yellow occurs.

The student claims that methyl orange was the best choice for the indicator. Do you agree or disagree? Justify your answer.

The average bond length in $\text{P}_4$ molecules is 221 pm, whereas the average bond length in $\text{P}_2$ molecules is 189 pm. Explain why the average bond length in $\text{P}_4$ is greater than the average bond length in $\text{P}_2$.

A sample of $\text{P}_4(s)$ is placed in a sealed rigid container and heated to 1600. K, at which point the initial partial pressure of $\text{P}_4(g)$ is 0.470 atm. Some of the $\text{P}_4(g)$ decomposes into $\text{P}_2(g)$ as the system reaches equilibrium. At equilibrium, the partial pressure of $\text{P}_2(g)$ is determined to be 0.630 atm.

Calculate the value of the equilibrium constant, $K_p$, for the reaction at 1600. K. Show the work that leads to your answer.

The reaction is thermodynamically favorable under standard conditions only at temperatures above 1500 K. A student claims that the reaction must be endothermic because it is favorable only at high temperatures. Do you agree or disagree? Justify your answer using $\Delta S°_{rxn}$ and $\Delta G°_{rxn}$.

Which of the bonds in $\text{CBrClF}_2$ (C–Br, C–Cl, or C–F) is the most polar? Justify your answer.

In a molecule of $\text{CBrClF}_2$, the F–C–F bond angle is $106.8°$ and the Br–C–Cl bond angle is $112.3°$. Explain the difference in bond angles using principles of atomic structure and VSEPR theory.

Lewis diagrams and boiling points for $\text{CBr}_4$ and $\text{CBrClF}_2$ are given in the following table.

Clearly identify all the intermolecular forces in pure $\text{CBr}_4(l)$ and in pure $\text{CBrClF}_2(l)$.

$\text{CBr}_4(l)$:

$\text{CBrClF}_2(l)$:

In terms of the relative strengths of all the intermolecular forces, explain why pure $\text{CBr}_4(l)$ has a higher boiling point than pure $\text{CBrClF}_2(l)$ does.

The student measures an absorbance of 0.32 for a $\text{V}^{2+}(aq)$ solution. The concentration of $\text{V}^{2+}$ ions in the solution is represented by the particle diagram in the circle in Figure 2. In the circle in Figure 3, draw the correct number of $\text{V}^{2+}$ ions to represent the concentration of a solution with an absorbance of 0.08. (Water molecules have been omitted for clarity. Assume that each particle diagram represents the same volume.)

[Figure 2: A test tube with a magnified circle showing 7 dots labeled $\text{V}^{2+}$ ion, scattered inside the circle; captioned "Absorbance = 0.32".]

[Figure 3: A test tube with an empty magnified circle (student draws V²⁺ ions inside), legend "○ = V²⁺ ion"; captioned "Absorbance = 0.08".]

The student measures the absorbance of a different $\text{V}^{2+}(aq)$ solution with an unknown molarity, but the absorbance is higher than 0.36 (the upper limit of the graph). The student then transfers $3.00\ \text{mL}$ of this $\text{V}^{2+}(aq)$ solution to a $25.0\ \text{mL}$ volumetric flask, dilutes to the mark on the flask with distilled water, and measures the absorbance of the diluted solution.

The absorbance of the diluted solution is 0.22. Determine the molarity of $\text{V}^{2+}(aq)$ in the diluted solution.

Determine the molarity of $\text{V}^{2+}(aq)$ in the original, undiluted solution. Show the work that leads to your answer.

The concentration of $\text{V}^{2+}(aq)$ calculated in part B (ii) is lower than the actual concentration of the undiluted solution. The student claims that the calculated concentration is too low because during the dilution step, the final level of the solution was higher than the mark on the volumetric flask. Do you agree or disagree? Justify your answer.

Based on the information given, calculate the value of $\Delta H°_f$ for $\text{Na}_2\text{O}(s)$. Show the work that leads to your answer.

An $18.4\ \text{g}$ sample of $\text{Na}(s)$ reacts with $12.8\ \text{g}$ of $\text{O}_2(g)$ according to Equation 1. Calculate the total amount of heat, in kilojoules, released during this process. Show the work that leads to your answer.

Lattice enthalpy can be defined as the energy required to separate an ionic crystal into gaseous ions. $\text{Rb}_2\text{O}(s)$ and $\text{Na}_2\text{O}(s)$ have similar crystal structures, and their lattice enthalpies are given in Table 2.

Using Coulomb's law, explain why the lattice enthalpy of $\text{Rb}_2\text{O}(s)$ is smaller than that of $\text{Na}_2\text{O}(s)$.