- Identify the target equation for the formation of acetic acid.
- Reverse equation (1) and adjust equations (2) and (3) by multiplying by 2 to match the stoichiometric coefficients of the target equation.
- Sum the enthalpy changes of the adjusted reactions: \(\Delta H_f^{\circ} = 875 + 2(-394.51) + 2(-285.8)\).
- Conclude the total enthalpy change is \(\boxed{-485.62 \text{ kJ/mole}}\).
Explanation
- Analysis and Data Overview
The goal is to determine the standard heat of formation (\(\Delta H_f^{\circ}\)) for acetic acid (\(HC_2H_3O_2(l)\)). By definition, the heat of formation reaction involves forming one mole of the substance from its constituent elements in their standard states:
\[2C(s) + 2H_2(g) + O_2(g) \rightarrow HC_2H_3O_2(l)\]
We are provided with the following thermochemical equations:
1) \(HC_2H_3O_2(l) + 2O_2(g) \rightarrow 2CO_2(g) + 2H_2O(l)\), \(\Delta H_1 = -875 \text{ kJ/mole}\)
2) \(C(s) + O_2(g) \rightarrow CO_2(g)\), \(\Delta H_2 = -394.51 \text{ kJ/mole}\)
3) \(H_2(g) + \frac{1}{2} O_2(g) \rightarrow H_2O(l)\), \(\Delta H_3 = -285.8 \text{ kJ/mole}\)
Hess's Law allows us to combine these known reactions to obtain the target reaction enthalpy.
- Manipulating Thermochemical Equations
To build the target equation, we manipulate the given equations:
- Since \(HC_2H_3O_2(l)\) is a product in the target reaction but a reactant in equation (1), we reverse equation (1):
\(2CO_2(g) + 2H_2O(l) \rightarrow HC_2H_3O_2(l) + 2O_2(g)\), \(\Delta H_{1,rev} = -(-875) = 875 \text{ kJ/mole}\)
- To obtain the \(2C(s)\) required for the target, multiply equation (2) by 2:
\(2C(s) + 2O_2(g) \rightarrow 2CO_2(g)\), \(\Delta H_{2,adj} = 2 \times (-394.51) = -789.02 \text{ kJ/mole}\)
- To obtain the \(2H_2(g)\) required for the target, multiply equation (3) by 2:
\(2H_2(g) + O_2(g) \rightarrow 2H_2O(l)\), \(\Delta H_{3,adj} = 2 \times (-285.8) = -571.6 \text{ kJ/mole}\)
- Calculating the Heat of Formation
Summing the manipulated equations:
Reactants: \(2CO_2(g) + 2H_2O(l) + 2C(s) + 2O_2(g) + 2H_2(g) + O_2(g)\)
Products: \(HC_2H_3O_2(l) + 2O_2(g) + 2CO_2(g) + 2H_2O(l)\)
After canceling out the common species (\(2CO_2(g)\), \(2H_2O(l)\), and \(3O_2(g)\) from the reactant side and \(2O_2(g)\) from the product side), we are left with the target equation:
\(2C(s) + 2H_2(g) + O_2(g) \rightarrow HC_2H_3O_2(l)\)
Summing the enthalpy changes:
\(\Delta H_f^{\circ} = \Delta H_{1,rev} + \Delta H_{2,adj} + \Delta H_{3,adj}\)
\(\Delta H_f^{\circ} = 875 + (-789.02) + (-571.6) = -485.62 \text{ kJ/mole}\)
Thus, the standard heat of formation for acetic acid is \(-485.62 \text{ kJ/mole}\).
Examples
Understanding Hess's Law is essential in chemical engineering and industrial chemistry. For example, when designing an efficient production process for a chemical like acetic acid, engineers must calculate the overall energy consumption or heat release. By knowing the enthalpies of individual reaction steps, they can predict whether a process will be exothermic (releasing heat) or endothermic (absorbing heat), which is critical for maintaining safe operating temperatures and optimizing energy usage in large-scale reactors.