Factors affecting rate, catalysts and experimental design

Consider the heterogeneous reaction between solid calcium carbonate and an aqueous solution of hydrochloric acid: $\mathrm{CaCO_3(s) + 2\,HCl(aq) \rightarrow CaCl_2(aq) + CO_2(g) + H_2O(l)}$. To study its kinetics at $25\,^\circ\mathrm{C}$, two independent experiments are carried out, using in both exactly $5.0\,\mathrm{g}$ of $\mathrm{CaCO_3}$ and $100\,\mathrm{mL}$ of aqueous solution of $\mathrm{HCl}\ 1.0\,\mathrm{mol}\times\mathrm{L}^{-1}$. In the first experiment a single compact piece of marble is used and in the second, finely powdered marble.

1. a) Justify, using collision theory, in which of the two experiments the initial rate of gas evolution will be greater.
2. b) Explain, with reasoning, whether the total volume of $\mathrm{CO_2(g)}$ released at the end of the reaction will differ in the two cases.

The gas-phase synthesis of nitrogen dioxide takes place at $300\,^\circ\mathrm{C}$ according to the overall reaction: $\mathrm{2\,NO(g) + O_2(g) \rightarrow 2\,NO_2(g)}$. In a closed reactor fitted with a movable piston at $300\,^\circ\mathrm{C}$, a mixture is introduced initially containing $0.20\,\mathrm{mol}$ of $\mathrm{NO}$ and $0.10\,\mathrm{mol}$ of $\mathrm{O_2}$, occupying an initial volume $V_1$.

1. a) The experimental law is $v=k[\mathrm{NO}]^2[\mathrm{O_2}]$. State the partial orders and the overall order, and explain why these exponents require experimental evidence for an overall reaction.
2. b) Deduce quantitatively by what factor the initial reaction rate is multiplied if the piston is moved so that the volume of the reactor is halved ($V_2 = V_1 / 2$), keeping the temperature of the system constant.

The rate constant for the gas-phase thermal decomposition of ethanal, $\mathrm{CH_3CHO(g) \rightarrow CH_4(g) + CO(g)}$, has been measured experimentally at two different temperatures, giving the values $k_1 = 1.10\times 10^{-2}\,\mathrm{L}\times\mathrm{mol}^{-1}\times\mathrm{s}^{-1}$ at $700\,\mathrm{K}$ and $k_2 = 7.95\times 10^{-1}\,\mathrm{L}\times\mathrm{mol}^{-1}\times\mathrm{s}^{-1}$ at $800\,\mathrm{K}$. Assume that the Arrhenius equation applies, with a constant pre-exponential factor $A$ over this temperature range. Data: universal gas constant $R = 8,314\,\mathrm{J}\times\mathrm{mol}^{-1}\times\mathrm{K}^{-1}$.

1. a) Calculate the activation energy of the reaction in $\mathrm{kJ}\times\mathrm{mol}^{-1}$.
2. b) Determine the numerical value and the units of the pre-exponential factor $A$. You may use $k=A\exp(-E_a/(RT))$.

The thermal decomposition of a gaseous substance has an activation energy of $85.0\,\mathrm{kJ}\times\mathrm{mol}^{-1}$. When the reaction is carried out in the presence of a platinum mesh acting as a heterogeneous catalyst, the activation energy decreases to $52.0\,\mathrm{kJ}\times\mathrm{mol}^{-1}$. Assume that the pre-exponential factor $A$ in the Arrhenius equation remains identical in both processes and does not vary with temperature. Data: $R = 8,314\,\mathrm{J}\times\mathrm{mol}^{-1}\times\mathrm{K}^{-1}$.

1. a) Calculate the numerical factor ($k_{\mathrm{cat}} / k_{\mathrm{no\,cat}}$) by which the rate constant increases when the catalyst is used, if the reactor is kept at $298\,\mathrm{K}$.
2. b) Determine the absolute temperature to which the uncatalysed reaction would have to be heated for its rate constant to be exactly equal to that of the catalysed reaction at $298\,\mathrm{K}$. You may use $k=A\exp(-E_a/(RT))$.

In the industrial synthesis of sulfur trioxide by the contact process, sulfur dioxide is oxidised at $450\,^\circ\mathrm{C}$ through an exothermic equilibrium: $\mathrm{2\,SO_2(g) + O_2(g) \rightleftharpoons 2\,SO_3(g)}$, with $\Delta H^\circ = -198\,\mathrm{kJ}\times\mathrm{mol}^{-1}$. To increase the rate of the process, solid divanadium pentoxide, $\mathrm{V_2O_5(s)}$, is used as a catalyst.

1. a) Explain how $\mathrm{V_2O_5(s)}$ modifies the activation energy of the forward reaction and of the reverse reaction.
2. b) Explain whether the introduction of the catalyst increases the standard enthalpy of the reaction $\Delta H^\circ$ or the concentration of $\mathrm{SO_3(g)}$ reached at chemical equilibrium at that temperature.

The attached figure compares the potential energy profiles against the reaction coordinate for the same gas-phase transformation carried out by an uncatalysed pathway and by a pathway with a homogeneous catalyst:

. In this graph, the enthalpy level of the reactants is at $20\,\mathrm{kJ}\times\mathrm{mol}^{-1}$ and that of the products at $0\,\mathrm{kJ}\times\mathrm{mol}^{-1}$. The peak of the uncatalysed profile reaches $100\,\mathrm{kJ}\times\mathrm{mol}^{-1}$, whereas that of the catalysed process lies at $60\,\mathrm{kJ}\times\mathrm{mol}^{-1}$.

1. a) Determine analytically the activation energies of the forward and reverse reactions for the uncatalysed pathway and for the catalysed pathway.
2. b) Calculate the enthalpy change of the reaction ($\Delta H$) and explain whether the catalyst changes the position of the chemical equilibrium.

The decomposition of hydrogen peroxide, $\mathrm{2\,H_2O_2(aq) \rightarrow 2\,H_2O(l) + O_2(g)}$, can be accelerated by the enzyme catalase (a biocatalyst) or by solid manganese dioxide, $\mathrm{MnO_2(s)}$ (an inorganic catalyst). In a series of trials carried out between $10\,^\circ\mathrm{C}$ and $70\,^\circ\mathrm{C}$ it is observed that, with $\mathrm{MnO_2}$, the reaction rate increases continuously as the temperature rises. However, with catalase, the rate reaches a maximum value at around $37\,^\circ\mathrm{C}$ and falls sharply until it disappears completely at $65\,^\circ\mathrm{C}$.

1. a) Explain at the molecular level why the enzyme loses its activity at high temperatures, in contrast to the inorganic catalyst.
2. b) Define the concept of enzyme specificity and state one fundamental advantage of using enzymes in chemical and industrial synthesis.

You are to design a safe laboratory experiment to determine quantitatively how the concentration of a reactant affects the initial rate of the reaction: $\mathrm{CaCO_3(s) + 2\,HCl(aq) \rightarrow CaCl_2(aq) + H_2O(l) + CO_2(g)}$. The available equipment is marble in homogeneous chips, aqueous solution of $\mathrm{HCl}\ 1.0\,\mathrm{mol}\times\mathrm{L}^{-1}$, distilled water, $100\,\mathrm{mL}$ Erlenmeyer flasks, a $50\,\mathrm{mL}$ gas syringe with an airtight connection, a stopwatch, measuring cylinders, a precision balance and a thermostatic bath at $25\,^\circ\mathrm{C}$.

1. a) Describe the experimental design with reasoning, identifying precisely the independent variable, the dependent variable and at least three essential control variables that must be kept strictly constant.
2. b) Explain the analytical procedure for calculating the initial rate and state the relevant safety precautions for this experiment. The design is to be a proposal for a supervised school laboratory: size the quantities so as not to exceed the capacity of the syringe and avoid blocking its movement.

Topic challenge. How would you optimise the speed of action of an effervescent medicine? Design and carry out a home kinetic study with effervescent tablets (or bicarbonate of soda) and water:

1. Experimental phase: Use your phone's stopwatch to measure the time ($t$) for complete reaction in four trials: a whole tablet in cold water (check the temperature of your fridge or a thermometer, $\approx 5.0\text{ }^\circ\text{C}$), whole at room temperature (record the one from Met Éireann for your locality or an indoor thermometer, $\approx 20.0\text{ }^\circ\text{C}$), crushed at room temperature, and whole with a few drops of vinegar added.

2. Data processing: Construct a table with your measurements and calculate the relative mean rate ($v_m \propto 1/t$). Justify the changes observed according to collision theory and the activation energy ($E_a$).

3. Modelling and mechanism: Propose a rate law ($v = k[\text{A}]^\alpha$), consistent with your mechanism, justifying the order of reaction with respect to the reactants (your trials do not vary concentrations, so they cannot determine it experimentally), and formulate a hypothetical two-step mechanism in which you identify the rate-determining step and discuss the role of the water and the acid (solvent, reactant or catalyst).

Submit a brief one-page report with your tables, calculations and conclusions applied to the pharmaceutical field.

Assessment criteria:

• Rigour in the home experimental design and precision in recording your own data.

• Coherent explanation of the kinetic factors using collision theory and $E_a$.

• Appropriate physico-chemical justification of the rate law and of the proposed step-by-step mechanism.