Question

Consider the reaction$$\begin{aligned}\mathrm{H}_{2}(g)+\mathrm{Cl}_{2}(g) \longrightarrow 2 \mathrm{HCl}(g) & \\\\\Delta H=-184.6 \mathrm{~kJ} / \mathrm{mol} \end{aligned}$$If 3 moles of \(\mathrm{H}_{2}\) react with 3 moles of \(\mathrm{Cl}_{2}\) to form \(\mathrm{HCl}\) calculate the work done (in joules) against a pressure of \(1.0 \mathrm{~atm} .\) What is \(\Delta U\) for this reaction? Assume the reaction goes to completion and that \(\Delta V=0\). \((1 \mathrm{~L} \cdot \mathrm{atm}=101.3 \mathrm{~J})\)

Short answer

The work done is 0 J, and \( \Delta U = -553800 \) J.

Step-by-step solution

Verify Reaction Composition

Check the balanced chemical equation. The reaction is \( \mathrm{H}_{2}(g) + \mathrm{Cl}_{2}(g) \longrightarrow 2 \mathrm{HCl}(g) \). From this equation, we can see that 1 mole of \( \mathrm{H}_{2} \) reacts with 1 mole of \( \mathrm{Cl}_{2} \) to produce 2 moles of \( \mathrm{HCl} \).

Calculate Moles of Products Formed

Since 3 moles of \( \mathrm{H}_{2} \) react with 3 moles of \( \mathrm{Cl}_{2} \), the number of moles of \( \mathrm{HCl} \) produced is 2 times the number of moles of \( \mathrm{H}_{2} \) or \( \mathrm{Cl}_{2} \). Thus, 6 moles of \( \mathrm{HCl} \) will be produced.

Calculate Change in Internal Energy (\( \Delta U \))

The change in internal energy \( \Delta U \) can be calculated using the formula \( \Delta U = \Delta H - P\Delta V \). Given \( \Delta V = 0 \), the equation simplifies to \( \Delta U = \Delta H \). For this reaction \( \Delta H = -184.6 \) kJ/mol, and since 3 moles of the reaction occur, \( \Delta H_{\text{total}} = 3 \times (-184.6) = -553.8 \) kJ. Thus, \( \Delta U = -553.8 \) kJ.

Calculate Work Done (\( w \))

The work done by a reaction is given by \( w = -P\Delta V \), where \( P \) is pressure and \( \Delta V \) is volume change. Since \( \Delta V = 0 \), \( w = 0 \) regardless of the pressure. Thus, the work done is 0 J.

Convert \( \Delta U \) to Joules

Since we have \( \Delta U = -553.8 \) kJ, we convert it to joules by using the conversion factor \(1\) kJ = \(1000\) J. Therefore, \( \Delta U = -553.8 \times 1000 = -553800 \) J.

Key concepts

Enthalpy

Enthalpy is a central concept in thermochemistry, representing the total heat content of a system. It is denoted by the symbol \( \Delta H \), and changes in enthalpy occur during chemical reactions. Enthalpy change is measured in kilojoules per mole (kJ/mol) and indicates whether a reaction is endothermic or exothermic. In the given reaction, \( \Delta H = -184.6 \) kJ/mol, meaning the process is exothermic, releasing heat to the surroundings. This negative value of \( \Delta H \) helps us verify that energy is released when hydrogen and chlorine form hydrogen chloride (HCl).
It is essential to grasp that enthalpy change provides critical insight into a reaction's heat evolution and guides predictions about temperature changes in the system. Thus, knowing the enthalpy change allows chemists to predict how much heat energy will be released or absorbed during a reaction, which is vital in designing reactions for industrial or laboratory purposes.

Internal Energy

Internal energy is another critical concept in thermochemistry that reflects the total energy contained within a system. The change in internal energy \( \Delta U \) is related to changes in heat and work. In this exercise, the relationship is simplified using the formula: \[ \Delta U = \Delta H - P\Delta V \]However, since \( \Delta V = 0 \), the change in volume term drops out, simplifying it to \( \Delta U = \Delta H \).
This means the change in internal energy is directly equal to the change in enthalpy, indicating that no work is done by the system (e.g., expansion or contraction). The total internal energy change for three moles reacting is \(-553.8 \text{ kJ}\), or \(-553800 \text{ J}\) when converted to joules. This conversion is crucial in aligning our understandings with energy values typically used in scientific calculations.
Internal energy includes all kinetic and potential energies of molecules within the system, hence it is integral for understanding how energy is conserved/transformed within chemical reactions.

Chemical Reactions

Chemical reactions involve the rearrangement of atoms to form new substances and can be characterized by chemical equations. In our exercise, the reaction between hydrogen gas \((\text{H}_2)\) and chlorine gas \((\text{Cl}_2)\) forms hydrogen chloride gas \((\text{HCl})\). This type of reaction is called a synthesis reaction, where two reactants combine to form a more complex product.
The stoichiometry of the reaction details that one mole of \(\text{H}_2\) reacts with one mole of \(\text{Cl}_2\) to produce two moles of \(\text{HCl}\). The balanced chemical equation helps us ensure that the conservation of mass is respected in the reaction, providing a theoretical basis to calculate the quantities of reactants/products involved.
Chemical reactions are influenced by thermodynamic quantities like enthalpy and internal energy, allowing scientists to understand energy flow, predict reaction feasibility, and design applications effectively.

Work in Thermodynamics

Work is an important concept in thermodynamics that reflects energy transfer that occurs when a force moves an object. In thermochemistry, it typically relates to the volume changes against external pressure during reactions. The equation for work done, \( w = -P\Delta V \), shows that work is associated with volume changes at pressure \( P \).
In the exercise, \( \Delta V \) is zero, implying there is no physical expansion or contraction of the reaction mixture leading to no work done, meaning \( w = 0 \). This simplifies calculations as the energy changes from volume work are non-existent, allowing us to focus on heat change (enthalpy) to determine internal energy changes.
Understanding work in the context of thermodynamics allows us to comprehend how energy is transferred in reaction systems, especially when gas reactions are involved, where volume change is likely.