Question

Calcium will reduce \(\mathrm{MgO}(\mathrm{s})\) to \(\mathrm{Mg}(\mathrm{s})\) at all temperatures from 0 to \(2000^{\circ} \mathrm{C}\). Use this fact, together with the melting point ( \(839^{\circ} \mathrm{C}\) ) and boiling point \(\left(1484^{\circ} \mathrm{C}\right)\) of calcium, to sketch a plausible graph of \(\Delta G^{\circ}\) as a function of temperature for the reaction \(2 \mathrm{Ca}(\mathrm{s})+\) \(\mathrm{O}_{2}(\mathrm{g}) \longrightarrow 2 \mathrm{CaO}(\mathrm{s})\).

Short answer

The graph begins with \(\Delta G^{\circ}\) at a negative value at 0°C. There are two points where \(\Delta G^{\circ}\) is zero, which are at the melting point (839°C) and boiling point (1484°C) of Calcium. Otherwise, the \(\Delta G^{\circ}\) is negative from 0 to 2000°C, indicating that the reaction is spontaneous over this entire temperature range.

Step-by-step solution

Identify Reaction Spontaneity

Given that Calcium reduces Magnesium Oxide to Magnesium at all temperatures from 0 to 2000°C, it means the reaction is spontaneous at this temperature range. Hence, \(\Delta G^{\circ}\) is negative throughout this range.

Consider the Phase Changes of Calcium

Calcium has a melting point of \(839^{\circ} \mathrm{C}\) and a boiling point of \(1484^{\circ} \mathrm{C}\). At these temperatures, \(\Delta G^{\circ}\) is zero as phase changes are equilibrium processes where Gibbs free energy change is zero.

Sketch the \(\Delta G^{\circ}\) vs Temperature Graph

The graph starts with negative \(\Delta G^{\circ}\) at 0°C indicating spontaneous reaction, it then moves to zero at \(839^{\circ} \mathrm{C}\) due to melting point, goes negative again as temperature increases, then to zero at boiling point \(1484^{\circ} \mathrm{C}\), and then again goes negative showing the reaction is still spontaneous at temperature above boiling point of Calcium.

Key concepts

Thermodynamics

Thermodynamics is the branch of physics that deals with heat, work, and energy. It's essential to understand how different types of energy transfer and transform. One of the main ideas in thermodynamics is the concept of Gibbs Free Energy (\( \Delta G \)). This tells us whether a process will occur spontaneously. In simpler terms, it helps us understand if a reaction can happen on its own without outside help. Gibbs Free Energy combines enthalpy (\( H \)), temperature (\( T \)), and entropy (\( S \)) into a single value. It's calculated with the formula:\[ \Delta G = \Delta H - T\Delta S \]- **Enthalpy (\( \Delta H \))**: This represents the heat absorbed or released during a reaction.- **Entropy (\( \Delta S \))**: This describes the disorder or randomness in a system.- **Temperature (\( T \))**: It is the measure of the heat in a system measured in Kelvin.In thermodynamics, a negative \( \Delta G \) value means the reaction can occur spontaneously, releasing energy. Positive \( \Delta G \) values mean it needs energy to proceed. By understanding these principles, you can predict how reactions will behave under different conditions.

Spontaneity of Reactions

Spontaneity in chemical reactions hints at whether a reaction can occur on its own. If a reaction is spontaneous, it means it can proceed without needing an external energy source. The Gibbs Free Energy (\( \Delta G \)) is an essential tool to verify this. Here's what the values indicate:

• **Negative \( \Delta G \)**: This implies a spontaneous reaction. The system releases energy, making it favorable at the given temperature.
• **Zero \( \Delta G \)**: The system is at equilibrium. There is no net change as the forces driving the reaction balance out.
• **Positive \( \Delta G \)**: This shows non-spontaneity. The reaction needs outside energy to occur.

Looking at the original exercise, since calcium reduces magnesium oxide at temperatures up to 2000°C, \( \Delta G \) remains negative, signaling spontaneity. Even at phase transition points like the melting and boiling of calcium, the general negativity of \( \Delta G \) across the temperature range confirms that the reaction is favorable.

Phase Transitions

Phase transitions are changes between solid, liquid, and gas phases. Understanding these transitions is vital when examining reactions over broad temperature ranges. Phase transitions are crucial in determining reactions' spontaneity and energy behavior. When substances switch phases at their melting or boiling points, there's a temporary equilibrium in energy terms. This means that the Gibbs Free Energy (\( \Delta G \)) at these points is zero. The system does not release or absorb extra energy overall because the change is balanced. For calcium:

• At **839°C**, calcium transitions from solid to liquid. The \( \Delta G \) value touches zero, indicating equilibrium.
• At **1484°C**, calcium changes from liquid to vapor. Again \( \Delta G \) equals zero, showing a balance during this transition.

After these phases, the reaction might revert to being spontaneous, depending on surrounding conditions. Therefore, understanding these transitions helps predict the shifts in reaction spontaneity as temperature changes.