- Identify that the target reaction is the sum of the two provided chemical equations.
- Recognize that adding the equations cancels the intermediate species \(\text{H}_2\text{O}(g)\).
- Apply Hess's Law to calculate the total enthalpy: \(\Delta H_{\text{rxn}} = \Delta H_1 + \Delta H_2\).
- Calculate the final value: \(-802 \text{ kJ} + (-88 \text{ kJ}) = \boxed{-890 \text{ kJ}}\).
Explanation
- Analyzing the Given Reactions and Objective
We are given two thermochemical equations representing steps in a chemical process:
1) \(\text{CH}_4(g) + 2\text{O}_2(g) \rightarrow \text{CO}_2(g) + 2\text{H}_2\text{O}(g)\) with \(\Delta H_1 = -802 \text{ kJ}\).
2) \(2\text{H}_2\text{O}(g) \rightarrow 2\text{H}_2\text{O}(l)\) with \(\Delta H_2 = -88 \text{ kJ}\).
Our objective is to find the enthalpy change \(\Delta H_{\text{rxn}}\) for the target reaction:
\(\text{CH}_4(g) + 2\text{O}_2(g) \rightarrow \text{CO}_2(g) + 2\text{H}_2\text{O}(l)\).
Hess's Law states that the total enthalpy change for a net chemical reaction is the sum of the enthalpy changes of the individual steps that constitute the reaction. If the target reaction can be formed by adding or subtracting the given steps, the overall enthalpy change is found by applying the same algebraic operations to their enthalpy values.
- Applying Hess's Law
To determine the target reaction, we examine the reactants and products of the provided equations. We notice that the target reaction features \(\text{CH}_4(g)\) and \(\text{O}_2(g)\) as reactants, and \(\text{CO}_2(g)\) and \(\text{H}_2\text{O}(l)\) as products.
Comparing this to our given steps:
- Step (1) contains the reactants \(\text{CH}_4(g)\) and \(2\text{O}_2(g)\) and the product \(\text{CO}_2(g)\).
- Step (2) contains the reactant \(2\text{H}_2\text{O}(g)\) and the product \(2\text{H}_2\text{O}(l)\).
When we add reaction (1) and reaction (2), the intermediate species \(2\text{H}_2\text{O}(g)\) appears on both sides of the combined equation:
\(\text{CH}_4(g) + 2\text{O}_2(g) + 2\text{H}_2\text{O}(g) \rightarrow \text{CO}_2(g) + 2\text{H}_2\text{O}(g) + 2\text{H}_2\text{O}(l)\).
Canceling the \(2\text{H}_2\text{O}(g)\) from both sides leaves us exactly with our target reaction:
\(\text{CH}_4(g) + 2\text{O}_2(g) \rightarrow \text{CO}_2(g) + 2\text{H}_2\text{O}(l)\).
Therefore, the target enthalpy change is simply the sum of the enthalpy changes of the two given equations: \(\Delta H_{\text{rxn}} = \Delta H_1 + \Delta H_2\).
- Final Calculation and Conclusion
Now, we perform the final numerical calculation for the enthalpy change:
\(\Delta H_{\text{rxn}} = \Delta H_1 + \Delta H_2 = -802 \text{ kJ} + (-88 \text{ kJ})\).
Calculating this sum gives \(\Delta H_{\text{rxn}} = -890 \text{ kJ}\).
Thus, the overall reaction equation showing the calculation is \(\Delta H_{\text{rxn}} = \Delta H_1 + \Delta H_2\), and the total enthalpy change is \(\boxed{-890 \text{ kJ}}\).
Examples
Hess's Law is essential in chemistry for determining the heat of reaction for processes that might be difficult to measure directly in a lab. For instance, if you were designing a heating system using a combustion reaction, you would need the exact energy output. Even if you cannot measure the enthalpy of a complete reaction in a single step, you can break it down into measurable intermediate steps, add their enthalpies using Hess's Law, and calculate the total energy, ensuring the safety and efficiency of your design.