Question

Given the following data $$ \begin{aligned} \mathrm{Ca}(s)+2 \mathrm{C}(\text { graphite }) & \longrightarrow \mathrm{CaC}_{2}(s) & \Delta H &=-62.8 \mathrm{~kJ} \\ \mathrm{Ca}(s)+\frac{1}{2} \mathrm{O}_{2}(g) & \longrightarrow \mathrm{CaO}(s) & \Delta H &=-635.5 \mathrm{~kJ} \\ \mathrm{CaO}(s)+\mathrm{H}_{2} \mathrm{O}(l) & \longrightarrow \mathrm{Ca}(\mathrm{OH})_{2}(a q) & \Delta H &=-653.1 \mathrm{~kJ} \\ \mathrm{C}_{2} \mathrm{H}_{2}(g)+\frac{5}{2} \mathrm{O}_{2}(g) & \longrightarrow 2 \mathrm{CO}_{2}(g)+\mathrm{H}_{2} \mathrm{O}(l) & \Delta H &=-1300 . \mathrm{kJ} \\ \mathrm{C}(\text { graphite })+\mathrm{O}_{2}(g) & \Delta H &=-393.5 \mathrm{~kJ} \end{aligned} $$ calculate \(\Delta H\) for the reaction $$ \mathrm{CaC}_{2}(s)+2 \mathrm{H}_{2} \mathrm{O}(l) \longrightarrow \mathrm{Ca}(\mathrm{OH})_{2}(a q)+\mathrm{C}_{2} \mathrm{H}_{2}(g) $$

Short answer

The enthalpy change for the reaction \( \mathrm{CaC}_{2}(s)+2 \mathrm{H}_{2} \mathrm{O}(l) \longrightarrow \mathrm{Ca}(\mathrm{OH})_{2}(a q)+\mathrm{C}_{2} \mathrm{H}_{2}(g) \) is \(-569.6 ~ \mathrm{kJ}\).

Step-by-step solution

Identify the reactions to be used

We need to use the given reactions to form the desired reaction. We can see that we need reactions (2) and (3) to form Ca(OH)\(_2\)(aq). Also, we need reaction (1) to include CaC\(_2\)(s) and H\(_2\)O(l) from reaction (4). The last term in the desired reaction is C\(_2\)H\(_2\)(g) which comes from reaction (4).

Adjust the reactions according to the desired reaction

Here are the adjustments we need to make to the given reactions: 1. Reverse reaction (1) to have CaC\(_2\)(s) on the left side. 2. Reverse reaction (3) to have Ca(OH)\(_2\)(aq) on the right side. 3. Divide reaction (4) by 2, as we only need C\(_2\)H\(_2\)(g) and not 2CO\(_2\)(g) on the right side. This leads to the following adjusted reactions: 1. \( \mathrm{CaC}_{2}(s) \longrightarrow \mathrm{Ca}(s)+2 \mathrm{C}(\text { graphite }) ~~~~~~~ \Delta H = 62.8 \mathrm{~kJ} \) 2. \( \mathrm{Ca}(s)+\frac{1}{2} \mathrm{O}_{2}(g) \longrightarrow \mathrm{CaO}(s) ~~~~~~~~~~~~ \Delta H = -635.5 \mathrm{~kJ} \) 3. \( \mathrm{Ca}(\mathrm{OH})_{2}(a q) \longrightarrow \mathrm{CaO}(s)+\mathrm{H}_{2} \mathrm{{O}}(l)~~~~ \Delta H = 653.1 \mathrm{~kJ} \) 4. \( \mathrm{C}_{2} \mathrm{H}_{2}(g)+\frac{5}{4} \mathrm{O}_{2}(g) \longrightarrow \mathrm{CO}_{2}(g) + \mathrm{H}_{2} \mathrm{O}(l)~~~~~~~~~~ \Delta H = -650 . \mathrm{kJ} \)

Add the adjusted reactions

Now, we will add the adjusted reactions together: \( \begin{aligned} &\mathrm{CaC}_{2}(s) &\longrightarrow &\mathrm{Ca}(s)+2 \mathrm{C}(\text { graphite }) &+62.8 \mathrm{~kJ} \\ +&\mathrm{Ca}(s)+\frac{1}{2} \mathrm{O}_{2}(g) &\longrightarrow &\mathrm{CaO}(s) &-635.5 \mathrm{~kJ} \\ +&\mathrm{Ca}(\mathrm{OH})_{2}(a q) &\longrightarrow &\mathrm{CaO}(s)+\mathrm{H}_{2} \mathrm{O}(l) &+653.1 \mathrm{~kJ} \\ +&\mathrm{C}_{2} \mathrm{H}_{2}(g)+\frac{5}{4} \mathrm{O}_{2}(g) &\longrightarrow &\mathrm{CO}_{2}(g) + \mathrm{H}_{2} \mathrm{O}(l) &-650 . \mathrm{kJ} \\ \hline &\mathrm{CaC}_{2}(s)+2 \mathrm{H}_{2} \mathrm{O}(l) &\longrightarrow &\mathrm{Ca}(\mathrm{OH})_{2}(a q)+\mathrm{C}_{2} \mathrm{H}_{2}(g) \end{aligned} \) Since we have the reaction we are looking for, we will now add the \(\Delta H\) values together: $$ \Delta H = 62.8 - 635.5 + 653.1 - 650 = -569.6 ~ \mathrm{kJ} $$

State the final answer

The enthalpy change for the reaction is: $$ \Delta H(\mathrm{CaC}_{2}(s)+2 \mathrm{H}_{2} \mathrm{O}(l) \longrightarrow \mathrm{Ca}(\mathrm{OH})_{2}(a q)+\mathrm{C}_{2} \mathrm{H}_{2}(g)) = -569.6 ~ \mathrm{kJ} $$

Key concepts

Hess's law

Understanding Hess's law is essential for calculating the enthalpy change for a reaction when direct measurement isn't feasible. It states that if a reaction is carried out in a series of steps, the total enthalpy change for the reaction equals the sum of the enthalpy changes for each individual step. This is because enthalpy is a state function, meaning it depends only on the initial and final states, not on the path taken. Hess's law allows us to manipulate known reactions (flip, multiply or divide them) to add up to a desired reaction, and then do the same with their enthalpy changes to find the overall enthapy change.

For example, if you wish to determine the enthalpy change of a complex reaction, you could break it down into simpler steps, find or measure the enthalpy changes for these known steps, and apply Hess's law to calculate the total enthalpy change. This approach is particularly valuable when direct measurement of the enthalpy change for a reaction is impractical or impossible.

Enthalpy of reaction

The enthalpy of reaction, or reaction enthalpy, refers to the heat energy absorbed or released during a chemical reaction at constant pressure. It is usually expressed in units of kilojoules per mole (kJ/mol). A negative enthalpy change (Delta H) indicates an exothermic reaction, where heat is emitted to the surroundings. Conversely, a positive Delta H indicates an endothermic reaction, where heat is taken in from the surroundings.

In solving enthalpy problems, it's crucial to correctly interpret the signs and magnitudes of the enthalpy changes provided for each reaction. Aligning the given reactions to match the target reaction ensures an accurate calculation of the overall enthalpy change by adding the enthalpy changes of the aligned steps, taking signs into account.

Chemical thermodynamics

Chemical thermodynamics studies the interrelation of heat and work with chemical reactions or with physical changes of state within the confines of the laws of thermodynamics. It encompasses concepts like enthalpy, entropy, and Gibbs free energy. Enthalpy is just one part of a larger puzzle, reflecting the heat change under constant pressure but not considering the disorder (entropy) or the spontaneity (Gibbs free energy) of a reaction.

In the context of enthalpy change calculations, thermodynamics provides the framework for understanding how energy is conserved and transferred during chemical processes. Knowing the principles of thermodynamics enables us to predict whether a reaction will occur spontaneously and to calculate the balance of heat and work involved in chemical reactions.

Stoichiometry

Stoichiometry is the quantitative relationship between reactants and products in a chemical reaction. It's based on the conservation of mass and the principle that matter cannot be created or destroyed. Using stoichiometry, chemists can predict the amount of products formed from given reactants or the amount of reactants needed to create a certain amount of product.

In calculating enthalpy changes, stoichiometry helps ensure that the amounts of reactants and products are properly balanced. When adjusting reactions to match a target reaction using Hess's law, stoichiometric principles guide us on how to multiply, divide, or reverse reactions, ensuring that the coefficients reflect the actual amounts of substances involved. This accuracy is crucial for the correct addition of individual enthalpy changes to yield the total enthalpy change of the overall reaction.