Textbook Problem

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“Heater Meals” are food packages that contain their own heat source, lust pour water into the heater unit, wait a few minutes, and voila! You have a hot meal.

Mg(s) + 2 H₂O(ℓ) → Mg(OH)₂(s) + H₂(g)

1. (a) Confirm that this is a product-favored reaction at equilibrium at 25 °C.
2. (b) What mass of magnesium is required to produce sufficient energy to heat 225 mL of water (density = 0.995 g/mL) from 25 °C to the boiling point?

Interpretation Introduction

Interpretation:

It should be predicted that if the given reaction of magnesium with water is product-favoured.

Concept introduction:

The Gibbs free energy or the free energy change is a thermodynamic quantity represented by ${\text{Δ}}_{\text{r}}{\text{G}}^{\text{o}}$. It is related to entropy and entropy by the following expression,

  ${\text{Δ}}_{\text{r}}{\text{G}}^{\text{o}}{\text{=Δ}}_{\text{r}}{\text{H}}^{\text{o}}{\text{-TΔ}}_{\text{r}}{\text{S}}^{\text{o}}$

It can be calculated in a similar manner as entropy and enthalpy.  The expression for the free energy change is:

  ${\text{Δ}}_{\text{r}}{\text{G}}^{\text{°}}\text{= }\sum {\text{nΔ}}_{\text{f}}{\text{G}}^{\text{°}}\left(\text{products}\right)\text{- }\sum {\text{nΔ}}_{\text{f}}{\text{G}}^{\text{°}}\left(\text{reactants}\right)$

The sign of ${\text{Δ}}_{\text{r}}{\text{G}}^{\text{o}}$ should be positive for a product-favored reaction. Thus, spontaneous reactions are referred to those that have negative free energy formation.

The heat $\text{q}$ is related to the specific heat by the expression,

  $\text{q = -mcΔT}$

Here, $\text{ΔT}$ is the temperature change and $\text{m}$ is the mass of the substance. $\text{c}$ is the specific heat which is constant. The specific heat of water is $4.186\text{ J/g }°\text{C}$.

Explanation of Solution

The value of ${\text{ΔG}}^{\text{o}}$ for the reaction of magnesium with water is calculated below.

Given:

Refer to Appendix L for the values of standard free energy values.

The given reaction is,

  $\text{Mg}\left(\text{s}\right){\text{ + 2H}}_{\text{2}}\text{O}\left(\text{l}\right)\to \text{Mg}{\left(\text{OH}\right)}_{\text{2}}\left(\text{s}\right)+{\text{H}}_{\text{2}}\left(\text{g}\right)$

The ${\text{Δ}}_{\text{r}}{\text{G}}^{\text{°}}$ for $\text{Mg}{\left(\text{OH}\right)}_{\text{2}}\left(\text{s}\right)$ is $-\text{833}\text{.51 kJ/mol}$.

The ${\text{Δ}}_{\text{r}}{\text{G}}^{\text{°}}$ for ${\text{H}}_{\text{2}}\left(\text{g}\right)$ is $\text{0 kJ/mol}$.

The ${\text{Δ}}_{\text{r}}{\text{G}}^{\text{°}}$ for $\text{Mg}\left(\text{s}\right)$ is $\text{0 kJ/mol}$.

The ${\text{Δ}}_{\text{r}}{\text{G}}^{\text{°}}$ for ${\text{H}}_{\text{2}}\text{O}\left(\text{l}\right)$ is $-\text{237}\text{.15 kJ/mol}$.

$\begin{array}{c}{\text{Δ}}_{\text{r}}{\text{G}}^{\text{°}}\text{= }\sum {\text{nΔ}}_{\text{f}}{\text{G}}^{\text{°}}\left(\text{products}\right)\text{- }\sum {\text{nΔ}}_{\text{f}}{\text{G}}^{\text{°}}\left(\text{reactants}\right)\\ \text{=}[\begin{array}{l}[\begin{array}{l}\left(\text{1 mol Mg}{\left(\text{OH}\right)}_{\text{2}}\left(\text{s}\right)\text{/mol-rxn}\right){\text{Δ}}_{\text{f}}{\text{G}}^{\text{°}}\left[\text{Mg}{\left(\text{OH}\right)}_{\text{2}}\left(\text{s}\right)\right]\text{+}\\ ({\text{1 mol}}_{}\end{array}\end{array}\end{array}$

Interpretation Introduction

Interpretation:

The mass of magnesium needed to produce energy in order to heat $\text{225mL}$ from given temperature to its boiling point should be identified.

Concept introduction:

The Gibbs free energy or the free energy change is a thermodynamic quantity represented by ${\text{Δ}}_{\text{r}}{\text{G}}^{\text{o}}$. It is related to entropy and entropy by the following expression,

  ${\text{Δ}}_{\text{r}}{\text{G}}^{\text{o}}{\text{=Δ}}_{\text{r}}{\text{H}}^{\text{o}}{\text{-TΔ}}_{\text{r}}{\text{S}}^{\text{o}}$

It can be calculated in a similar manner as entropy and enthalpy.  The expression for the free energy change is:

  ${\text{Δ}}_{\text{r}}{\text{G}}^{\text{°}}\text{= }\sum {\text{nΔ}}_{\text{f}}{\text{G}}^{\text{°}}\left(\text{products}\right)\text{- }\sum {\text{nΔ}}_{\text{f}}{\text{G}}^{\text{°}}\left(\text{reactants}\right)$

The sign of ${\text{Δ}}_{\text{r}}{\text{G}}^{\text{o}}$ should be positive for a product-favored reaction. Thus, spontaneous reactions are referred to those that have negative free energy formation.

The heat $\text{q}$ is related to the specific heat by the expression,

  $\text{q = -mcΔT}$

Here, $\text{ΔT}$ is the temperature change and $\text{m}$ is the mass of the substance. $\text{c}$ is the specific heat which is constant. The specific heat of water is $4.186\text{ J/g }°\text{C}$.