Understanding Enthalpy of Formation

The enthalpy of formation ($\Delta H_f^\circ$) is a fundamental concept in thermochemistry. It is defined as the change in enthalpy that accompanies the formation of one mole of a substance from its constituent elements in their standard states, under standard conditions (usually 298.15 K and 1 bar pressure).

A key point to remember is the convention for the enthalpy of formation of elements in their standard states. By definition, the standard enthalpy of formation ($\Delta H_f^\circ$) of an element in its most stable form under standard conditions is taken as zero.

Let's examine what the standard states are for the elements relevant to the options provided:

• Oxygen (O): The standard state is diatomic oxygen gas, $O_2(g)$.
• Copper (Cu): The standard state is solid copper, $Cu(s)$.
• Hydrogen (H): The standard state is diatomic hydrogen gas, $H_2(g)$.

Analyzing the Options for Non-Zero Enthalpy of Formation

We are asked to identify the substance for which the enthalpy of formation is non-zero. Let's look at each option based on the definition of enthalpy of formation and standard states:

1. $O_2$

   Oxygen ($O_2$) is the standard state of the element oxygen under standard conditions. According to the definition, the enthalpy of formation of an element in its standard state is zero. Therefore, the enthalpy of formation of $O_2$ is zero.

2. $Cu$

   Copper ($Cu$) exists as a solid metal in its standard state under standard conditions. It is the standard state of the element copper. Therefore, the enthalpy of formation of $Cu$ is zero.

3. $O_3$

   Ozone ($O_3$) is an allotrope of oxygen. However, the standard state of oxygen is $O_2$, not $O_3$. Ozone is formed from $O_2$. The formation reaction is $3/2 \, O_2(g) \rightarrow O_3(g)$. Since $O_3$ is not the standard state of the element oxygen, its enthalpy of formation is non-zero. It requires energy to convert $O_2$ into $O_3$.

4. $H^+$

   Hydrogen ion ($H^+$) is typically considered in aqueous solution ($H^+(aq)$). By convention, the standard enthalpy of formation of $H^+(aq)$ is defined as zero. This is a reference point for tabulating enthalpies of formation of other ions in solution. While the process of forming $H^+$ from $H_2$ involves ionization and bond breaking, the standard enthalpy of formation for $H^+(aq)$ is assigned a value of zero for practical reasons in calculating thermodynamic properties of ions.

Comparing the options, $O_2$ and $Cu$ are elements in their standard states, so their enthalpy of formation is zero. $H^+(aq)$ has an enthalpy of formation defined as zero by convention. $O_3$ is an allotrope of oxygen and is not the standard state of the element; its formation from $O_2$ has a definite enthalpy change.

Substance with Non-Zero Enthalpy of Formation

Based on the analysis, Ozone ($O_3$) is the substance among the options whose standard enthalpy of formation is non-zero. This is because it is not the standard state of the element oxygen.

Summary of Enthalpy of Formation ($\Delta H_f^\circ$) for the given options:

Therefore, the enthalpy of formation is nonzero for $O_3$. Understanding the concept of standard states and how they relate to the definition of enthalpy of formation is crucial for solving such problems in thermochemistry.