Question

Hydrogen has been suggested as the fuel of the future. One way to store it is to convert it to a compound that can be heated to release the hydrogen. One such compound is calcium hydride, \(\mathrm{CaH}_{2}\). This compound has a heat of formation of \(-186.2 \mathrm{~kJ} / \mathrm{mol}\) and a standard entropy of \(42.0 \mathrm{~J} / \mathrm{mol} \cdot \mathrm{K}\). What is the minimum temperature to which calcium hydride would have to be heated to produce hydrogen at one atmosphere pressure?

Short answer

Answer: The minimum temperature required is approximately 4433 K.

Step-by-step solution

Write down the balanced chemical equation for the reaction

For the reaction of calcium hydride to produce hydrogen gas, the balanced chemical equation is: $$\mathrm{CaH}_{2(s)} \rightarrow \mathrm{Ca}_{(s)} + \mathrm{H}_{2(g)}$$

Calculate the Gibbs free energy change

The Gibbs free energy change for a reaction (\(\Delta G\)) can be calculated using the following equation: $$\Delta G = \Delta H - T\Delta S$$ where \(\Delta H\) is the enthalpy change (heat of formation), \(T\) is the temperature in Kelvin, and \(\Delta S\) is the entropy change. In this case, we are given that the heat of formation of calcium hydride is \(-186.2\mathrm{~kJ/mol}\) and the standard entropy is \(42.0\mathrm{~J/mol\cdot K}\). Since we want to find the minimum temperature required to produce hydrogen gas, we will set the Gibbs free energy change to zero and solve for \(T\). When \(\Delta G = 0\), the reaction is at equilibrium. $$0 = -186200\mathrm{~J/mol} - T (42.0\mathrm{~J/mol\cdot K})$$

Solve for the temperature

Rearrange the equation and solve for \(T\): $$T (42.0\mathrm{~J/mol\cdot K}) = 186200\mathrm{~J/mol}$$ $$T = \frac{186200\mathrm{~J/mol}}{42.0\mathrm{~J/mol\cdot K}}$$ $$T \approx 4433\mathrm{~K}$$ The minimum temperature to which calcium hydride would have to be heated to produce hydrogen at one atmosphere pressure is approximately \(4433\mathrm{~K}\).

Key concepts

Enthalpy Change

Enthalpy change, denoted as \(\Delta H\), is a measure of the total energy change during a chemical reaction in terms of heat under constant pressure. It's a state function, which fundamentally means that its value depends only on the initial and final states of the system, not on the path taken to get from one to the other. In the context of the exercise, the enthalpy change is the heat of formation of calcium hydride, which is \(\Delta H = -186.2 \text{ kJ/mol}\). This negative value indicates that the formation of calcium hydride from its elements releases energy to the surroundings, and thus, it is an exothermic reaction.

Understanding \(\Delta H\) is crucial because it allows us to predict whether a reaction would require heat from the surroundings or release heat. If \(\Delta H\) is negative, as it is in the formation of calcium hydride, the substance can potentially serve as a fuel, releasing energy as it forms more stable compounds. Knowing the enthalpy change is also essential to calculate the Gibbs free energy change, which determines the spontaneity of a reaction.

Standard Entropy

Standard entropy, symbolized by \(S^\circ\), is the absolute entropy of a substance at the standard state of 1 bar of pressure. It quantifies the amount of disorder or randomness within a chemical system. Standard entropy is typically given in units of joules per mole per kelvin (\text{J/mol\cdot K}). In the given exercise, we are provided with the standard entropy of calcium hydride as \(S^\circ = 42.0 \text{ J/mol\cdot K}\).

The concept of entropy is significant in predicting the feasibility of a reaction under given conditions. For reactions where the standard entropy increases (\(\Delta S > 0\)), the system becomes more disordered, which is generally favored by nature. In the exercise, we are interested in the change in entropy (\(\Delta S\)) rather than the standard entropy, which is considered along with the enthalpy change when using the Gibbs free energy equation to determine reaction spontaneity. Understanding entropy changes helps in estimating how temperature affects the direction and extent of a chemical reaction.

Chemical Reaction Equilibrium

Chemical reaction equilibrium occurs when a reaction and its reverse reaction proceed at the same rate, leading to stable concentrations of the reactants and products. When a system is at equilibrium, the Gibbs free energy change, \(\Delta G\), is equal to zero. This is a state of balance where no net change occurs in the concentrations of reactants and products. It is a dynamic equilibrium, meaning that the reactions are still occurring, but there's no observable change in the system.

In the exercise, setting the Gibbs free energy change to zero and solving for the temperature gives us the minimum temperature needed to produce hydrogen gas from calcium hydride at one atmosphere of pressure. The condition where \(\Delta G = 0\) and \(T\) is solved represents the point at which the substances involved are at equilibrium under those specific conditions. This concept is vital in industrial chemistry and process engineering because it dictates the conditions necessary for maximizing the yield of a desired chemical product.