Rate and mechanism of the reaction of Mg and HCl

Question

In the reaction between hydrochloric acid and Magnesium the overall reaction is:

$$\ce{Mg + 2HCl -> MgCl2 + H2}$$

My proposed mechanism of reaction is that $\ce{HCl}$ dissociates in $\ce{H2O}$ like so: $$\ce{H2O + HCl <=> H3O+ + Cl- }$$

And then the $\ce{H3O+}$ oxidises the magnesium like so:

$$\ce{2H3O+ + Mg -> Mg^{2+} + H2O + H2}$$

This, in an experiment to determine the rate of the reaction, measuring the $\ce{H2}$ produced would measure this mechanism, because it is the step in which hydrogen gas is evolved. Presumably these Magnesium ions are then in solution. I would therefore deduce that the step: $$\ce{2H3O+ + Mg -> Mg^{2+} + H2O + H2}$$ Is the rate-determining step, because it is the step in which magnesium ions are formed and go into solution. My rate law, deduced from these mechanisms, would look like this:

$$\ce{rate = k * [H3O+] }$$ Because it is the concentration of the $\ce{H3O+}$ which determines the formation of the Mg ions. From there, I would assume, the $\ce{Mg^{2+}}$ and the $\ce{Cl-}$ would remain in solution, as the reaction is carried out in the acid, obviously consisting of water.

From this, can I say that the rate law for the reaction

$$\ce{Mg + 2HCl -> MgCl2 + H2}$$

is merely: $$\ce{rate = k * [H3O+] }$$ But $\ce{[H3O+]}$ is: $$\ce{[H3O+] = (K*[H2O][HCl])/[Cl^{-}] }$$ Where K is the equilibrium constant of the acid equilibrium.

So can we write our rate law like:

$$\ce{rate = k*(K[H2O][HCl])/[Cl^{-}] }$$ Where K is the equilibrium constant and k is the rate constant and might look like this (following the Arrhenius equation): $$\ce{k = A e^{-E_a/(R T)}}$$

To make our rate law like this: $$\ce{ rate = A e^{-E_a/(R T)} *((K*[H2O][HCl])/[Cl^{-}]) }$$

My question is: Is my rate law correct for this reaction? If not, where have I gone wrong?

Additionally: This was one of our practicals. We ended up concluding that rate goes up quadratically or exponentially with concentration. This rate law would seem to indicate a linear increase, not exponential. I assume in my rate law there is no contribution from the solid Mg. Obviously in a powder, more surface area is exposed, and a high rate is experienced. Is there a way to factor this into my rate law? Thanks for the help! :)

Answer 1

Your rate law should be $$ r = k_\text{obs} \times [\ce{H3O+}]^2 = A_\text{obs} \exp\left(\frac{E_\text{a, obs}}{RT}\right) \times [\ce{H3O+}]^2\; ,$$ which nicely fits your experimental data of a quadratic dependence on the concentration for the rate.

The exponent is necessary because the reaction is seemingly second-order in the concentration of $\ce{H+}$.

Answer 2

For the reaction $\ce{2H3O+ + Mg -> H2 + Mg^{-2}}$; the actual rate law is $r=k \times [\ce{H3O+}]^2 \times (\ce{Mg})^1.$

The most obvious error is that HCl is a strong acid, thus not in equilibrium. Cl- is merely a spectator ion.

A more accurate reaction step method is:

$\ce{H3O+ + Mg -> H2O + MgH}$

$\ce{H3O+ + MgH -> H2 + Mg^{-2}}$

The surface area of magnesium directly affects the k value, but magnesium is still a valid part of the reaction. The magnesium term may appear zeroth order if the magnesium is in far excess, so significantly lowering the initial input of magnesium will probably result in the correct rate law.