Question

\(\Lambda\) mixturc of 2 moles of \(C O\) and 1 mole of \(\mathrm{O}_{2}\) in a closed vessel is ignited to convert to \(\mathrm{CO}\) into \(\mathrm{CO}_{2}\) then (1) \(\Delta H=\Delta U\) (2) \(\Delta H>\Delta U\) (3) \(\Delta H<\Delta U\) (4) The relationship depends upon the capacity of the vessel

Short answer

ΔH < ΔU

Step-by-step solution

- Reaction Analysis

Identify the chemical reaction taking place. The reaction is: \[ 2 CO(g) + O_2(g) \rightarrow 2 CO_2(g) \]

- Calculate the Change in Gases Moles

In the given reaction, the initial moles of gas are 2 moles of CO and 1 mole of O2, making a total of 3 moles. The final product, CO2, has 2 moles. Therefore, the change in moles of gas is \[ \text{Δn} = \text{final moles} - \text{initial moles} = 2 - 3 = -1 \]

- Apply Thermodynamic Relationship

Recall the relationship between ΔH and ΔU: \[ \text{ΔH} = \text{ΔU} + \text{Δn}RT \] where Δn is the change in moles of gas, R is the gas constant, and T is the temperature.

- Interpret the Result

Given \[ \text{Δn} = -1 \], the equation becomes \[ \text{ΔH} = \text{ΔU} - RT \]. This means ΔH is less than ΔU because a negative value is being subtracted from ΔU.

- Conclusion

Based on the analysis, ΔH is less than ΔU for the given reaction. Therefore, the correct answer is: \[ \text{(3) ΔH < ΔU} \]

Key concepts

enthalpy change

Enthalpy change, symbolized as \(\text{ΔH}\), is a crucial concept in thermodynamics. It represents the heat exchange in a chemical reaction at constant pressure. When we discuss enthalpy, we are focusing on the heat flow into or out of the system during the reaction.
Enthalpy change is particularly useful for understanding energy changes during chemical reactions in open systems or systems at constant pressure. For example:

• In an exothermic reaction, heat is released, and \(\text{ΔH}\) is negative.
• In an endothermic reaction, heat is absorbed, and \(\text{ΔH}\) is positive.

In the given exercise, the enthalpy change is influenced by the number of gas moles changing during the reaction.
This alters the overall energy exchange due to the creation or consumption of gaseous molecules.
Understanding \(\text{ΔH}\) helps predict how much energy will be exchanged in the form of heat during chemical reactions.

internal energy change

Internal energy change, denoted as \(\text{ΔU}\), refers to the total change in a system's internal energy. This encompasses all kinetic and potential energies of the molecules within the system. Unlike enthalpy, internal energy change is not restricted to constant pressure processes.
Internal energy change is the sum of heat added to the system and the work done on it. Mathematically, it's expressed as \(\text{ΔU} = \text{q} + \text{w}\), where \(\text{q}\) is heat and \(\text{w}\) is work.
In the context of chemical reactions:

• When a system absorbs heat and no work is done, \(\text{ΔU}\) increases.
• When work is done on the system and no heat is exchanged, \(\text{ΔU}\) also increases.

The given reaction involves gases, hence the internal energy change can also be impacted by changes in pressure and volume. The relationship between \(\text{ΔH}\) and \(\text{ΔU}\) in the exercise is explored via the equation \(\text{ΔH} = \text{ΔU} + \text{ΔnRT}\), highlighting the effects of gas mole changes and temperature.

chemical reactions

Chemical reactions involve the transformation of reactants into products through the breaking and forming of chemical bonds. This process has associated energy changes that can be analyzed using thermodynamic properties.
In the given exercise, the reaction is:
\[2 CO(g) + O_2(g) \rightarrow 2 CO_2(g)\]
Here, carbon monoxide and oxygen react to form carbon dioxide. The exercise evaluates the thermodynamic changes during this process by analyzing the change in the number of gas moles (\text{Δn}).
Understanding chemical reactions involves:

• Identifying reactants and products.
• Counting the moles of reactants and products.
• Applying thermodynamic equations to determine energy changes.

This helps determine properties such as enthalpy change (\text{ΔH}) and internal energy change (\text{ΔU}). In the given exercise, the relationship \[ \text{ΔH} = \text{ΔU} + \text{Δn}RT \] was used to identify that \(\text{ΔH}\) is less than \(\text{ΔU}\), due to the reduction in gas moles resulting in energy release by the system.