Mathematics: notes and practice questions

A student investigates electrochemical cells using the following standard electrode potentials.

(i) Identify the species that is the strongest oxidizing agent in the table.

(ii) Identify the species that is the strongest reducing agent in the table.

A voltaic cell is constructed using the zinc and copper half-cells under standard conditions.

(i) State the half-equation for the reaction occurring at the anode.

(ii) Calculate the standard cell potential, $E^\ominus_{cell}$, in V.

Deduce, justifying your answer, whether a spontaneous reaction will occur when solid silver metal, $Ag(s)$, is placed into a $1.00 \text{ mol dm}^{-3}$ solution of iron(II) nitrate.

A student investigates the decomposition of hydrogen peroxide, H₂O₂, in a commercial antiseptic solution when exposed to light over a period of 7 days. The reaction is:

$2\text{H}_2\text{O}_2(\text{aq}) \rightarrow 2\text{H}_2\text{O}(\text{l}) + \text{O}_2(\text{g})$

The concentration of H₂O₂ is determined by titration with a standard solution of acidified potassium manganate(VII), KMnO₄. Several identical flasks are prepared. One flask is kept in darkness, while others are exposed to a light source for different durations each day.

(a) Identify two variables, other than the volume of the solution, that should be controlled in this experiment.

(b) (i) The antiseptic solution is diluted with deionized water before titration. Suggest why this is necessary.

(b) (ii) Identify a possible systematic error associated with the control sample kept in darkness.

(b) (iii) Suggest how the experimental setup could be improved to check if the systematic error identified in (b)(ii) is significant.

(c) The following data are collected for one of the titrations:

Final burette reading = $24.50 \pm 0.05~\text{cm}^3$

Initial burette reading = $2.30 \pm 0.05~\text{cm}^3$

Calculate the percentage uncertainty of the titre.

(d) The initial concentration of H₂O₂ in all flasks was $0.88~\text{mol}~\text{dm}^{-3}$. The concentration after 7 days was measured for each flask.

(i) Calculate the average rate of decrease in hydrogen peroxide concentration, in $\text{mol}~\text{dm}^{-3}~\text{day}^{-1}$, for the sample exposed to light for 4 hours daily over the 7-day period.

(d) (ii) The student's hypothesis is: "A greater decrease in hydrogen peroxide concentration will be observed in solutions exposed to light for longer durations, due to photochemical decomposition."

Discuss, with reference to the data, the extent to which the results support this hypothesis.

(d) (iii) State one implication of the results for the storage of hydrogen peroxide solutions.

(e) Suggest a relevant extension to this investigation that would provide further information on the stability of hydrogen peroxide solutions.

A student investigated the effectiveness of three different brands of antacid tablets (Brand A, Brand B, and Brand C) in neutralizing hydrochloric acid, which simulates stomach acid. The active ingredient in all three brands is calcium carbonate, $CaCO_3$.

The following procedure was used for each brand:

1. One tablet was crushed using a mortar and pestle.

2. The crushed tablet was transferred to a conical flask containing $50.00 \text{ cm}^3$ of $0.500 \text{ mol dm}^{-3}$ HCl(aq). This is an excess of acid.

3. The mixture was stirred for a fixed time until the reaction appeared complete.

4. The resulting solution was titrated with $0.250 \text{ mol dm}^{-3}$ NaOH(aq) using a suitable indicator.

5. The experiment was repeated three times for each brand.

(a) State the independent and dependent variables for this investigation.

(b) Suggest two experimental conditions, other than the volumes and concentrations of the acid and alkali, that must be controlled to ensure a fair comparison between the brands.

(c) Suggest a reason for crushing the tablet before adding it to the acid.

The average results for the titrations are shown in the table.

(d) Based on the data, deduce which brand of antacid is the most effective, giving a reason.

(e) Calculate the amount, in mol, of HCl neutralized by one tablet of Brand B.

(f) The manufacturer of Brand A claims their tablet contains $0.900 \text{ g}$ of calcium carbonate, $CaCO_3$. Determine if the student's results for Brand A support this claim. The equation for the reaction is: $CaCO_3(s) + 2HCl(aq) \rightarrow CaCl_2(aq) + H_2O(l) + CO_2(g)$.

Antimony (Sb) and Bismuth (Bi) are elements in group 15 of the periodic table.

(a) Antimony has two stable isotopes. 57.21% of antimony atoms contain 70 neutrons and the remainder contain 72 neutrons.

(i) Deduce the nuclear symbol of the isotope of antimony containing 72 neutrons. Use section 6 of the data booklet.

(ii) Calculate, to two decimal places, the relative atomic mass of antimony.

Bismuth(III) nitrate, $Bi(NO_3)_3$, is a common salt of bismuth.

(b) (i) The compound contains both ionic and covalent bonds. State which particles are joined by covalent bonds and which are joined by ionic bonds.

(ii) Distinguish between covalent and ionic bonding in terms of electron distribution.

(iii) State the enthalpy term that characterizes the strength of the bonding between the ions in an ionic solid.

(iv) Write an equation for the formation of aqueous bismuth(III) nitrate from solid bismuth(III) oxide and nitric acid.

(v) Calculate the volume, in $cm^3$, of $1.50 \text{ mol dm}^{-3}$ nitric acid required to react completely with $5.00 \text{ g}$ of solid bismuth(III) oxide.

(vi) Predict, with a reason, whether bismuth(III) oxide is expected to be primarily acidic, basic or amphoteric.

(vii) Discuss how the relative reactivity of zinc and bismuth could be established using the metals and aqueous solutions of their nitrates.

(viii) Discuss the products formed at the electrodes during the electrolysis of aqueous bismuth(III) nitrate. Use the standard electrode potential $E^\ominus(Bi^{3+}/Bi) = +0.32 \text{ V}$ and section 24 of the data booklet.

Bismuth compounds are sometimes used in fireworks to produce special effects.

(c) (i) State the feature of the atomic emission spectrum of an element that corresponds to its first ionization energy.

(ii) Calculate the wavelength, in nm, that corresponds to the first ionization energy of bismuth. Use sections 1, 2 and 8 of the data booklet.

(iii) Explain why the first ionization energy of bismuth is lower than that of polonium (Po), in terms of nuclear charge and electron shielding.

Strontium is an alkaline earth metal in group 2 of the periodic table, used to produce the red colour in fireworks.

(a) In a sample of strontium, 10.0% of the atoms have a mass number of 86 and the remainder have a mass number of 88.

(i) Deduce the nuclear symbol of the isotope of strontium with mass number 86. Use section 6 of the data booklet.

(ii) Calculate, to two decimal places, the relative atomic mass of this sample of strontium.

(iii) Explain why the first ionization energy of strontium is greater than that of rubidium.

(b) Strontium nitrate has the formula $Sr(NO_3)_2$.

(i) The compound contains both ionic and covalent bonds. State which particles are joined by covalent bonds and which are joined by ionic bonds.

(ii) Contrast covalent and ionic bonds in terms of how the valence electrons are involved.

(iii) Write a balanced chemical equation for the formation of strontium nitrate solution from the reaction of solid strontium hydroxide with nitric acid.

(iv) Calculate the volume, in $cm^3$, of $1.50 \text{ mol dm}^{-3}$ nitric acid required to react completely with $5.00 \text{ g}$ of solid strontium hydroxide.

(v) Predict, with a reason, whether strontium oxide is acidic, basic or amphoteric.

(vi) Describe an experiment to establish the relative reactivity of strontium and zinc using the metals and aqueous solutions of their nitrates.

A food chemist is analyzing the components of a new artificial sweetener mixture using paper chromatography. A small spot of the mixture is applied to a chromatography paper, and the paper is placed in a sealed chamber with a solvent system consisting of $70\%$ ethanol and $30\%$ water. After some time, the solvent front moves up the paper, separating the components.

(a) Define the stationary phase and the mobile phase in this paper chromatography experiment.

(b) Explain why it is important that the initial spot of the sweetener mixture is applied above the solvent level in the chamber, and why the solvent front should not be allowed to reach the very top of the chromatography paper.

(c) After the chromatography run, the solvent front travelled $9.0\text{ cm}$ from the start line. Two distinct components, Sweetener A (yellow) and Sweetener B (blue), were observed to have travelled $7.2\text{ cm}$ and $4.5\text{ cm}$ respectively from the start line.

Calculate the retardation factor, $R_F$, for Sweetener A and Sweetener B. Show your working.

(d) The stationary phase (paper) is polar due to cellulose, and the mobile phase is a moderately polar mixture of $70\%$ ethanol and $30\%$ water.

Deduce, with a reason, which sweetener (A or B) is more polar.

(e) If Sweetener A and another component had very similar $R_F$ values and were not well separated, suggest one modification to the experiment that could improve their separation.

A student plans to determine the concentration of copper(II) ions in a sample of industrial wastewater using colorimetry. The blue colour of the solution is due to the hydrated copper(II) ion, $[Cu(H_2O)_6]^{2+}$. To do this, they must first prepare a standard solution of copper(II) sulfate and construct a calibration curve.

(a) Describe the procedure for preparing a standard aqueous solution of known concentration from solid hydrated copper(II) sulfate, $CuSO_4 \cdot 5H_2O(s)$.

(b) Describe how this standard solution can be used to construct a calibration curve.

(c) Explain how this curve can be used to determine the concentration of copper(II) ions in the wastewater sample.

Silicon, a key component in electronics, can be purified to an exceptional degree, making it one of the purest elements commercially available. It reacts with oxygen to form silicon dioxide, a compound with a vast range of applications from glassmaking to fibre optics.

(a) Distinguish between the terms element and compound.

(b) Justify, using electronegativity values from the data booklet, why the Si-O bond is polar.

(c) Contrast the electrical conductivity of silicon and silicon dioxide. Relate the difference to their bonding and structure.

(d) The first ionization of a gaseous silicon atom can be initiated by electromagnetic radiation. Calculate the wavelength, in m, of the radiation corresponding to the first ionization energy of silicon. Use the data booklet.

Sulfur dioxide, $SO_2(g)$, is a major air pollutant primarily released from the combustion of fossil fuels in power plants and other industrial facilities.

(a) Outline one reason why $SO_2$ is considered an atmospheric pollutant.

(b) A sample of flue gas from a coal-fired power plant has a volume of $5.00 \times 10^4 m^3$. The gas contains $0.0500\%$ $SO_2$ by volume at a temperature of $150 \degree C$ and a pressure of $1.01 \times 10^5 Pa$. Calculate the mass, in g, of $SO_2$ in this sample of flue gas.

Use sections 1, 2 and 6 of the data booklet.

(c) Explain why sulfur dioxide, $SO_2$, deviates more from ideal gas behaviour than carbon dioxide, $CO_2$.

(d) To reduce emissions, sulfur dioxide can be reacted with hydrogen sulfide, $H_2S$, to produce elemental sulfur.

(i) State the initial and final oxidation states of sulfur originating from sulfur dioxide in this reaction.

(ii) Deduce the second product of the reaction and write the balanced chemical equation.

A student is tasked with determining the concentration of copper(II) ions, $Cu^{2+}(aq)$, in a sample of industrial wastewater using a colorimeter. The blue colour of the solution is due to the hydrated copper(II) ion, $[Cu(H_2O)_6]^{2+}$. To do this, a calibration curve must first be prepared using a standard solution of hydrated copper(II) sulfate, $CuSO_4 \cdot 5H_2O(s)$.

(a) Describe the essential steps to prepare a standard aqueous solution of copper(II) sulfate of a known concentration.

(b) Describe how this standard solution can be used to construct a calibration curve.

(c) Explain how the calibration curve can be used to determine the concentration of $Cu^{2+}(aq)$ in the wastewater sample.

A voltaic cell is constructed using a copper half-cell, $Cu^{2+}(aq) + 2e^- \rightleftharpoons Cu(s)$.

(a) A solution of copper(II) ions is required for the half-cell. State the chemical formula of a suitable soluble copper(II) compound.

(b) The copper half-cell is connected to another half-cell to form a complete voltaic cell. A salt bridge is used to connect the two half-cells. Describe the function of the salt bridge.

(c) A high-resistance voltmeter is used to measure the potential difference of the cell. State the units of cell potential.

Sulfur dioxide, $SO_2(g)$, is a gas produced during the combustion of sulfur-containing fossil fuels, such as coal. It is a significant air pollutant.

(a) Outline a reason why $SO_2$ is a pollutant.

(b) Calculate the amount, in moles, of $SO_2$ in $5.00 \times 10^3 m^3$ of flue gas which contains $0.0500\%$ $SO_2$ by volume at $150 ^\circ C$ and $1.01 \times 10^5 Pa$. Use sections 1 and 2 of the data booklet.

(c) In a flue-gas desulfurization unit, $SO_2$ reacts with calcium carbonate and oxygen. The overall equation is:

$2CaCO_3(s) + 2SO_2(g) + O_2(g) \rightarrow 2CaSO_4(s) + 2CO_2(g)$

Deduce which element is oxidized in this process, indicating the initial and final oxidation states.

In an experiment to determine the formula of hydrated sodium sulfate, $\text{Na}_2\text{SO}_4\cdot x\text{H}_2\text{O}$, a sample was heated in a crucible to remove the water of crystallisation. The following data were recorded:

Mass of empty crucible $= 22.45\text{ g}$

Mass of crucible and hydrated sodium sulfate (before heating) $= 27.28\text{ g}$

Mass of crucible and anhydrous sodium sulfate (after heating) $= 24.58\text{ g}$

Describe what is meant by the phrase 'heated to constant mass'.

Determine the value of $x$ in the formula $\text{Na}_2\text{SO}_4\cdot x\text{H}_2\text{O}$ using the experimental data. Use section 6 of the data booklet.

The balance used to measure the masses has an uncertainty of $\pm 0.01\text{ g}$. Calculate the percentage uncertainty in the calculated mass of water lost.

The contents of the crucible were completely transferred to a $250\text{ cm}^3$ volumetric flask and dissolved in distilled water, making the solution up to the mark. Calculate the concentration, in $\text{mol dm}^{-3}$, of the sodium sulfate solution.

Calculate the mass of hydrated sodium sulfate, $\text{Na}_2\text{SO}_4\cdot 10\text{H}_2\text{O}$, that would be required to prepare $500\text{ cm}^3$ of a $0.100\text{ mol dm}^{-3}$ solution. Use section 6 of the data booklet.