Problem 20 - A Complete Resource Book in Chemistry for JEE Main

Question

Assume each reaction is carried out in an open container. For which reaction \(\Delta \mathrm{H}=\Delta \mathrm{U} ?\) (a) \(\mathrm{PCl}_{5}(\mathrm{~g}) \longrightarrow \mathrm{PCl}_{3}(\mathrm{~g})+\mathrm{Cl}_{2}(\mathrm{~g})\) (b) \(2 \mathrm{CO}(\mathrm{g})+\mathrm{O}_{2}(\mathrm{~g}) \longrightarrow 2 \mathrm{CO}_{2}(\mathrm{~g})\) (c) \(\mathrm{H}_{2}(\mathrm{~g})+\mathrm{Br}_{2}(\mathrm{~g}) \longrightarrow 2 \mathrm{HBr}(\mathrm{g})\) (d) \(\mathrm{C}(\mathrm{s})+2 \mathrm{H}_{2} \mathrm{O}(\mathrm{g}) \longrightarrow 2 \mathrm{H}_{2}(\mathrm{~g})+\mathrm{CO}_{2}(\mathrm{~g})\)

Step-by-Step Solution

Answer

For reaction (c), \(ΔH = ΔU\).

Step 1: Understanding the Relationship Between ΔH and ΔU

The enthalpy change (ΔH) and internal energy change (ΔU) of a reaction are related through the equation: \[ ΔH = ΔU + ΔnRT \] where Δn is the change in the number of moles of gas, R is the ideal gas constant, and T is the temperature in Kelvin. In an open container at constant pressure, if Δn = 0, then ΔH = ΔU.

Step 2: Analyzing Reaction (a)

The reaction is:\[ \mathrm{PCl}_{5} \rightarrow \mathrm{PCl}_{3} + \mathrm{Cl}_{2} \]The change in moles of gas (Δn) is calculated by subtracting the moles of reactants from products:Δn = (1 + 1) - 1 = 1. Since Δn ≠ 0, ΔH ≠ ΔU.

Step 3: Analyzing Reaction (b)

The reaction is: \[ 2 \mathrm{CO} + \mathrm{O}_{2} \rightarrow 2 \mathrm{CO}_{2} \]The change in moles of gas is:Δn = 2 - (2 + 1) = -1.Since Δn ≠ 0, ΔH ≠ ΔU.

Step 4: Analyzing Reaction (c)

The reaction is: \[ \mathrm{H}_{2} + \mathrm{Br}_{2} \rightarrow 2 \mathrm{HBr} \]The change in moles of gas is:Δn = 2 - (1 + 1) = 0.Since Δn = 0, ΔH = ΔU.

Step 5: Analyzing Reaction (d)

The reaction is: \[ \mathrm{C} + 2 \mathrm{H}_{2} \mathrm{O} \rightarrow 2 \mathrm{H}_{2} + \mathrm{CO}_{2} \]The change in moles of gas is:Δn = (2 + 1) - 2 = 1.Since Δn ≠ 0, ΔH ≠ ΔU.

Key Concepts

Enthalpy Change

Enthalpy change, symbolized as \(\Delta H\), refers to the amount of heat absorbed or released during a chemical reaction at constant pressure. This concept is crucial in understanding how energy interacts with matter when substances transform from reactants to products.
Chemically, \(\Delta H\) is related to the internal energy change \(\Delta U\) and is expressed as:\[ \Delta H = \Delta U + \Delta nRT \]Here, \(\Delta n\) represents the change in the number of moles of gaseous products minus reactants, \(R\) is the ideal gas constant, and \(T\) is the temperature in Kelvin. At constant pressure, if there is no change in the number of gas moles, then \(\Delta H\) equals \(\Delta U\).
Understanding enthalpy helps in predicting whether a reaction will release energy (exothermic) or absorb energy (endothermic). Additionally, it plays a critical role in processes like combustion and phase changes.

Ideal Gas Constant

The ideal gas constant, denoted as \(R\), is a fundamental component in the equation that relates enthalpy change to internal energy change. It serves as a vital link between macroscopic and microscopic worlds in thermodynamics, specifically in gas laws.
The value of \(R\) is approximately 8.314 J/mol·K. This constant appears in multiple equations, including the ideal gas law:\[ PV = nRT \]and the relationship for enthalpy change:\[ \Delta H = \Delta U + \Delta nRT \]In the context of chemical reactions, \(R\) helps account for the energy contributions from gas volume changes when a reaction occurs at constant pressure. This is especially relevant when determining the difference between \(\Delta H\) and \(\Delta U\), as \(R\) factors in the energy associated with the work done by gas expansion or compression at a given temperature.

Change in Moles of Gas

The change in moles of gas, \(\Delta n\), is a key factor that influences the relationship between enthalpy change and internal energy change. It represents the difference in the total number of moles of gaseous products and reactants in a chemical reaction.
Mathematically, \(\Delta n\) is expressed as:\[ \Delta n = \text{moles of gaseous products} - \text{moles of gaseous reactants} \]This value is significant in understanding whether a reaction will have an equal amount of heat exchange as energy change. If \(\Delta n = 0\), this implies that the sum of gaseous reactants equals the gaseous products, rendering \(\Delta H\) equal to \(\Delta U\). This often occurs when there is no net change in the gas volume under constant pressure conditions.
For reactions carried out in open systems, where pressure remains constant, \(\Delta n\) provides insight into the volumetric changes affected by the reaction and consequently the energy adjustment accounted by \(RT\).