Animated Solution for Chemistry - Chemical Thermodynamics: The standard heat of formation (${\Delta }_f{H}_{298}^{\circ }$) of ethane (in kJ/mol), if the heat of combustion of ethane, hydrogen and graphite are $-1560$, $-393.5$ and $-286\text{ kJ/mol}$, respectively is ………

Imagine you are an energy accountant, tasked with balancing the books of the universe. In the realm of chemical thermodynamics, energy cannot be created or destroyed; it can only change forms. This fundamental truth is the beating heart of Hess's Law, a principle that turns complex chemical reactions into elegant mathematical puzzles.

Hess's Law states that the total enthalpy change for a chemical reaction is independent of the pathway taken. Think of it like climbing a mountain. Whether you take the direct, steep path to the summit or a winding, scenic route, your final change in altitude is exactly the same. In thermodynamics, enthalpy is a state function, meaning it only cares about where you start (reactants) and where you end up (products).

In this problem, we are asked to find the standard heat of formation (${\Delta }_f{H}^{\circ }$) of ethane, ${\text{C}}_2{\text{H}}_6$. The direct path is the formation reaction from its constituent elements in their standard states:

However, we don't have the direct energy value for this path. Instead, we are given the heats of combustion for the reactants and the product. This allows us to construct an alternate route: combusting the reactants to form ${\text{CO}}_2$ and ${\text{H}}_2\text{O}$, and then conceptually "un-combusting" those products back into ethane.

By visualizing the Hess's Law cycle, we can derive a powerful master equation. The enthalpy of formation is simply the sum of the combustion enthalpies of the reactants minus the combustion enthalpy of the product.

Why do we subtract the product's combustion enthalpy? Because in our energy cycle, we are moving against the natural direction of the product's combustion. Reversing a reaction flips the sign of its enthalpy change.

Our master equation becomes:

Now, we must carefully substitute the given values. A common pitfall here is forgetting the stoichiometry. Enthalpy is an extensive property, meaning it depends on the amount of substance. Since our balanced equation requires 2 moles of carbon and 3 moles of hydrogen, we must multiply their respective molar combustion enthalpies by 2 and 3.

Plugging in the given values:

Let's execute the multiplication. Two times $-286$ is $-572$. Three times $-393.5$ is $-1180.5$. And subtracting a negative number is the same as adding a positive one, so $-(-1560)$ becomes $+1560$.

Adding the negative terms together gives us $-1752.5$. Finally, we add the $+1560$ to find our ultimate answer:

The negative sign is crucial. It tells us that the formation of ethane from graphite and hydrogen gas is an exothermic process, meaning it releases energy into the surroundings. By mastering Hess's Law, you unlock the ability to calculate the energy of almost any reaction, simply by finding the right alternate path!