Question

Calculate \(\Delta H\) for the reaction: $$ 2 \mathrm{NH}_{3}(g)+\frac{1}{2} \mathrm{O}_{2}(g) \longrightarrow \mathrm{N}_{2} \mathrm{H}_{4}(l)+\mathrm{H}_{2} \mathrm{O}(l) $$ given the following data: $$ 2 \mathrm{NH}_{3}(g)+3 \mathrm{N}_{2} \mathrm{O}(g) \longrightarrow 4 \mathrm{N}_{2}(g)+3 \mathrm{H}_{2} \mathrm{O}(l) $$ \(\Delta H=-1010 . \mathrm{kJ}\) $$ \mathrm{N}_{2} \mathrm{O}(g)+3 \mathrm{H}_{2}(g) \longrightarrow \mathrm{N}_{2} \mathrm{H}_{4}(l)+\mathrm{H}_{2} \mathrm{O}(l) $$ \(\Delta H=-317 \mathrm{kJ}\) $$ \mathrm{N}_{2} \mathrm{H}_{4}(l)+\mathrm{O}_{2}(g) \longrightarrow \mathrm{N}_{2}(g)+2 \mathrm{H}_{2} \mathrm{O}(l) $$ \(\Delta H=-623 \mathrm{kJ}\) $$ \mathrm{H}_{2}(g)+\frac{1}{2} \mathrm{O}_{2}(g) \longrightarrow \mathrm{H}_{2} \mathrm{O}(l) $$ \(\Delta H=-286 \mathrm{kJ}\)

Short answer

The enthalpy change for the target reaction is \(\Delta H = 393.5\, \text{kJ}.\)

Step-by-step solution

The target reaction we need to find \(\Delta H\) for is: $$ 2 \mathrm{NH}_{3}(g)+\frac{1}{2} \mathrm{O}_{2}(g) \longrightarrow \mathrm{N}_{2} \mathrm{H}_{4}(l)+\mathrm{H}_{2} \mathrm{O}(l) $$ #Step 2: Manipulate given reactions to match the target reaction#

We need to manipulate the given reactions in such a way that they can be added together to equal the target reaction. The following adjustments are made: a. In the first reaction, multiply by -2/3 to have -2 NH3(g) on the left side: $$ \frac{-2}{3}(2 \mathrm{NH}_{3}(g)+3 \mathrm{N}_{2} \mathrm{O}(g) \longrightarrow 4 \mathrm{N}_{2}(g)+3 \mathrm{H}_{2} \mathrm{O}(l)) $$ b. In the second reaction, reverse it to have the N2H4 (l) on the right side: $$ \mathrm{N}_{2} \mathrm{H}_{4}(l)+\mathrm{H}_{2} \mathrm{O}(l) \longleftrightarrow \mathrm{N}_{2} \mathrm{O}(g)+3 \mathrm{H}_{2}(g) $$ c. In the third reaction, multiply it by 1/2 to have 1/2 O2(g) on the left side: $$ \frac{1}{2}(\mathrm{N}_{2} \mathrm{H}_{4}(l)+\mathrm{O}_{2}(g) \longrightarrow \mathrm{N}_{2}(g)+2 \mathrm{H}_{2} \mathrm{O}(l)) $$ d. In the fourth reaction, no need to manipulate as it fits with the target reaction. #Step 3: Calculate the corresponding enthalpy changes for the adjusted reactions#

Use the manipulated equations to calculate the corresponding enthalpy changes: a. For the first equation, multiply the enthalpy by -2/3: $$ \Delta H_{1} = \frac{-2}{3}(-1010\,\text{kJ}) = 674\,\text{kJ} $$ b. For the second equation, reverse the reaction: $$ \Delta H_{2} = -(-317\,\text{kJ}) = 317\,\text{kJ} $$ c. For the third equation, multiply the enthalpy by 1/2: $$ \Delta H_{3} = \frac{1}{2}(-623\,\text{kJ}) = -311.5\,\text{kJ} $$ d. For the fourth equation, no manipulation is needed, so the enthalpy remains: $$ \Delta H_{4} = -286\,\text{kJ} $$ #Step 4: Add the adjusted reactions and their enthalpies to obtain the target reaction and enthalpy#

Now add the adjusted reactions and their respective enthalpies: Reactions: $$ \frac{-2}{3}(2 \mathrm{NH}_{3}(g)+3 \mathrm{N}_{2} \mathrm{O}(g))+( \mathrm{N}_{2} \mathrm{H}_{4}(l)+\mathrm{H}_{2} \mathrm{O}(l)+\frac{1}{2}(\mathrm{N}_{2} \mathrm{H}_{4}(l)+\mathrm{O}_{2}(g))+ (\mathrm{H}_{2}(g)+\frac{1}{2} \mathrm{O}_{2}(g)) = 2 \mathrm{NH}_{3}(g)+\frac{1}{2} \mathrm{O}_{2}(g) \longrightarrow \mathrm{N}_{2} \mathrm{H}_{4}(l)+\mathrm{H}_{2} \mathrm{O}(l) $$ Enthalpies: $$ \Delta H = \Delta H_{1} + \Delta H_{2} + \Delta H_{3} + \Delta H_{4} $$ $$ \Delta H = 674\,\text{kJ} + 317\,\text{kJ} - 311.5\,\text{kJ} - 286\,\text{kJ} = 393.5\,\text{kJ} $$ So, the enthalpy change for the target reaction is: $$ \Delta H = 393.5\,\text{kJ} $$

Key concepts

Understanding Hess's Law

Hess's Law is a principle used in thermochemistry to calculate enthalpy changes in chemical reactions. It states that the total enthalpy change of a reaction is the same, regardless of the pathway the reaction takes. This is because enthalpy is a state function, dependent only on the initial and final states of a system. This allows us to manipulate and combine chemical equations to find the enthalpy change (\(\Delta H\)) of a target reaction that may not be directly measurable. By rearranging given reactions and combining their enthalpy changes, we can determine the enthalpy change of a desired reaction.

Key Concepts in Thermochemistry

Thermochemistry is the branch of chemistry that deals with the study of heat and energy changes associated with chemical reactions. In this field, understanding the exchange of energy, especially in the form of heat, between systems is crucial. Thermochemistry relies on some core concepts:

• Enthalpy (\(H\)): This is the total heat content of a system and is a measure of energy in thermodynamic systems.
• Exothermic and Endothermic Reactions: Exothermic reactions release heat, resulting in a negative \(\Delta H\), while endothermic reactions absorb heat, leading to a positive \(\Delta H\).
• Heat of Reaction: This refers to the heat change associated with a chemical reaction, equivalent to the enthalpy change under constant pressure conditions.

Understanding these concepts helps to interpret and apply Hess's Law effectively.

Chemical Equations in Thermochemistry

Chemical equations represent chemical reactions, showing the reactants and products involved, as well as their respective coefficients. In thermochemistry, these equations are essential for understanding the stoichiometry and heat changes during reactions.
Manipulating these equations, as required in applying Hess's Law, involves:

• Balancing: Ensuring the number of atoms of each element is equal on both sides of the equation.
• Reversing Equations: Sometimes, we might need to reverse a reaction to cancel out unwanted products or reactants; this changes the sign of \(\Delta H\).
• Multiplying or Dividing: Adjusting coefficients by multiplying or dividing the entire reaction to achieve the desired stoichiometry affects the magnitude of \(\Delta H\) proportionally.

This ensures that the target reaction can be accurately constructed and analyzed.

Manipulation of Reactions

In thermochemistry, reaction manipulation is a critical skill used to apply Hess's Law for calculating unknown enthalpy changes. This involves altering the given reactions through specific mathematical operations:

• Reversing: Flip the direction of the reaction. The enthalpy change (\(\Delta H\)) must also be reversed in sign. This helps when a product in one reaction becomes a reactant in another.
• Scaling: Multiply or divide the stoichiometric coefficients to match the quantities needed. The enthalpy change is scaled by the same factor to maintain consistency in the energy account.
• Algebraic Addition: Add the manipulated reactions to achieve the target equation. The total enthalpy change (\(\Delta H\)) is the sum of the enthalpy changes from each individual reaction.

This methodical approach ensures that any pathway to the same chemical outcome has the same enthalpy change, allowing accurate determination of \(\Delta H\) for reactions that are otherwise difficult to measure directly.