Question

What is the enthalpy change for the unknown reaction? \(\mathrm{P}(\mathrm{s})+3 / 2 \mathrm{Br}_{2}(\ell) \rightarrow \operatorname{PBr} 3(\ell) \Delta H=-185 \mathrm{~kJ}\) \(\mathrm{PI}_{3}(\mathrm{~s}) \rightarrow \mathrm{P}(\mathrm{s})+3 / 2 \mathrm{I}_{2}(\mathrm{~s}) \Delta H=?\) $$ \mathrm{PI}_{3}(\mathrm{~s})+3 / 2 \mathrm{Br}(\ell) \rightarrow \mathrm{PBr}(\ell)+3 / 2 \mathrm{I}_{2}(\mathrm{~s}) \Delta H=-139 \mathrm{~kJ} $$

Short answer

The enthalpy change is 46 kJ.

Step-by-step solution

Write the Given Reactions

We have three reactions given:1. \( \mathrm{P}(\mathrm{s})+\frac{3}{2} \mathrm{Br}_{2}(\ell) \rightarrow \operatorname{PBr}_{3}(\ell) \) with \( \Delta H=-185 \mathrm{~kJ} \).2. \( \mathrm{PI}_{3}(\mathrm{s}) \rightarrow \mathrm{P}(\mathrm{s})+\frac{3}{2} \mathrm{I}_{2}(\mathrm{s}) \) with \( \Delta H=? \).3. \( \mathrm{PI}_{3}(\mathrm{s})+\frac{3}{2} \mathrm{Br}_{2}(\ell) \rightarrow\mathrm{PBr}_{3}(\ell)+\frac{3}{2} \mathrm{I}_{2}(\mathrm{s}) \) with \( \Delta H=-139 \mathrm{~kJ} \).

Understand Reaction Transformation

We need to determine the enthalpy change for the second reaction. Notice that the third reaction can be rewritten to demonstrate a transformation from \( \mathrm{PI}_{3} \) to \( \mathrm{PBr}_{3} \), which involves both the conversion of \( \mathrm{PI}_{3} \) to \( \mathrm{P} \) (the reverse of the second reaction) and the provided first reaction covering transformation \( \mathrm{P} \rightarrow \mathrm{PBr}_{3} \).

Apply Hess's Law

Hess's Law states that the total enthalpy change of a reaction is the same, no matter how many steps the reaction is carried out in. To find the enthalpy change for \( \mathrm{PI}_{3}(\mathrm{s}) \rightarrow \mathrm{P}(\mathrm{s})+\frac{3}{2} \mathrm{I}_{2}(\mathrm{s}) \):Using Hess's law,\(\Delta H_{3} = \Delta H_{2} + \Delta H_{1}\) where \( \Delta H_{3} = -139 \mathrm{~kJ} \) and \( \Delta H_{1} = -185 \mathrm{~kJ} \).\(-139 = \Delta H_{2} + (-185)\) which rearranges to:\(\Delta H_{2} = 185 - 139 = 46 \mathrm{~kJ}\).Thus, \( \Delta H_{2} \) is \( 46 \mathrm{~kJ} \).

Finalize Enthalpy Change

Hence, the enthalpy change for the reaction \( \mathrm{PI}_{3}(\mathrm{s}) \rightarrow \mathrm{P}(\mathrm{s})+\frac{3}{2} \mathrm{I}_{2}(\mathrm{s}) \) is \( 46 \mathrm{~kJ} \).

Key concepts

Hess's Law

Hess's Law is a crucial principle in thermodynamics. It tells us that the total enthalpy change ( \( \Delta H \) ) for a chemical reaction is the same, irrespective of whether it occurs in a single step or through multiple steps. This is incredibly useful when determining heats of reaction that are not easily measured directly. If you can break a complex reaction into simpler ones with known enthalpy changes, then by rearranging and summing these, you can calculate the overall energy change.

For example, if you have several reactions like:

• Reaction A: \( A \rightarrow B \) with \( \Delta H_1 \)
• Reaction B: \( B \rightarrow C \) with \( \Delta H_2 \)

Using Hess's Law, we know that the enthalpy change from \( A \) to \( C \) (\( \Delta H \)) equals \( \Delta H_1 + \Delta H_2 \). Hence, Hess's Law provides a way to calculate unknown enthalpy changes by using known values from different reactions.

In the exercise, we applied Hess's Law to discover the enthalpy change for the reaction \( \mathrm{PI}_{3}(\mathrm{s}) \rightarrow \mathrm{P}(\mathrm{s})+\frac{3}{2} \mathrm{I}_{2}(\mathrm{s}) \) by using given reactions with known changes.

Chemical Reactions

Chemical reactions involve the transformation of reactants into products. They encompass the breaking and forming of chemical bonds. Each reaction can either absorb energy (endothermic) or release energy (exothermic). The difference in energy between reactants and products is known as the enthalpy change ( \( \Delta H \) ).

In chemical equations, reactants are written on the left side and products on the right, often including the enthalpy change to indicate the energy involved. During any reaction:

• Exothermic reactions have a negative \( \Delta H \) because they release energy.
• Endothermic reactions have a positive \( \Delta H \) because they absorb energy.

The exercise explores chemical reactions involving phosphorus compounds like \( \mathrm{PBr}_{3} \) and \( \mathrm{PI}_{3} \), where the transformation includes energy changes that are calculated using Hess’s Law.

Thermodynamics

Thermodynamics is the branch of physics concerned with heat and temperature and their relation to energy and work. Within this field, one explores the principles governing chemical reactions and energy conversions. Regarding enthalpy, the first law of thermodynamics states that energy cannot be created or destroyed. Instead, it can only transform from one form to another.

Understanding enthalpy within thermodynamics allows scientists to predict whether a chemical reaction will occur spontaneously. Key principles include:

• Internal energy: the total energy within a system.
• Enthalpy (\( H \)): a measure of total heat content in a system.

The exercise illustrates these concepts through enthalpy changes during transformation reactions. Enthalpy calculations make use of the overarching thermodynamic principle of energy conservation.

Enthalpy Calculations

Enthalpy ( \( H \) ) is directly linked with heat that is either absorbed or released during chemical reactions at constant pressure. Calculating enthalpy changes enables chemists to understand the thermal aspect of reactions.

To determine the enthalpy change, follow these steps:

• Identify known enthalpy changes from related reactions.
• Use Hess’s Law to rearrange the known values to suit the desired reaction.
• Calculate the missing enthalpy using mathematical rearrangement, ensuring the laws of thermodynamics are satisfied.

For example, if given reactions have known \( \Delta H \), such as in our exercise, algebraic manipulation helps find unknown enthalpy changes. Through these calculations, the exercise found the enthalpy change for \( \mathrm{PI}_{3} \rightarrow \mathrm{P} \) and \( \mathrm{I}_{2} \).