Problem 110 - A Complete Resource Book in Chemistry for JEE Main

Question

For the reaction, \(\mathrm{A}(\mathrm{g})+2 \mathrm{~B}(\mathrm{~g}) \longrightarrow 2 \mathrm{C}(\mathrm{g})+3 \mathrm{D}(\mathrm{g})\) The value of \(\Delta \mathrm{H}\) at \(27^{\circ} \mathrm{C}\) is \(19.0 \mathrm{kcal}\). The value of \(\Delta \mathrm{E}\) for the reaction would be (given \(\mathrm{R}=2.0 \mathrm{cal} \mathrm{K}^{-\mathrm{t}} \mathrm{mol}^{-1}\) ) (a) \(20.8 \mathrm{kcal}\) (b) \(19.8 \mathrm{kcal}\) (c) \(18.8 \mathrm{kcal}\) (d) \(17.8 \mathrm{kcal}\)

Step-by-Step Solution

Answer

The value of \(9\mathrm{E}\) is 17.8 kcal (option d).

Step 1: Understand the Concept of Reaction Enthalpy and Internal Energy

The relationship between enthalpy change (9\mathrm{H}) and internal energy change (9\mathrm{E}) for reactions involving gases is given by the equation: \[\Delta \mathrm{H} = \Delta \mathrm{E} + \Delta n_gRT\] where 9\(n_g\) is the change in moles of gas, \(R\) is the gas constant, and \(T\) is the temperature in Kelvin.

Step 2: Calculate Change in Moles of Gas (9n_g)

From the balanced chemical equation \(\mathrm{A(g)} + 2 \mathrm{B(g)} \longrightarrow 2 \mathrm{C(g)} + 3 \mathrm{D(g)}\), calculate \(9n_g\):\[ \Delta n_g = (2 + 3) - (1 + 2) = 5 - 3 = 2 \] The change in moles of gas is 2.

Step 3: Convert Temperature to Kelvin

The given temperature is 279C. Convert it to Kelvin using the formula:\[T = 27 + 273 = 300 \text{ K}\]Therefore, the temperature is 300 K.

Step 4: Substitute Values into the Formula

Substitute the known values into the equation:\[\Delta \mathrm{H} = \Delta \mathrm{E} + \Delta n_gRT\]\[19.0 = \Delta \mathrm{E} + (2)(2.0)(300)\]Simplifying further gives:\[19.0 = \Delta \mathrm{E} + 1200/1000 \]Convert calories to kilocalories:\[19.0 = \Delta \mathrm{E} + 1.2 \]

Step 5: Solve for 9\mathrm{E}

Solve the equation to find 9\mathrm{E}:\[ \Delta \mathrm{E} = 19.0 - 1.2 = 17.8 \mathrm{kcal}\]The internal energy change for the reaction is 17.8 kcal.

Key Concepts

Enthalpy Change

Enthalpy change (
ackslashmathrm{H}) is an important concept in thermochemistry that measures the heat change during a chemical reaction at constant pressure. This value indicates whether a reaction absorbs or releases heat. If 
ackslashmathrm{H} is positive, the reaction is endothermic, meaning it absorbs heat. If it's negative, the reaction is exothermic, releasing heat into the surroundings.
In the given exercise, the enthalpy change is provided as 19.0 kcal at 27°C. This positive value indicates that the reaction is endothermic. It requires energy input from the environment to proceed.
Understanding enthalpy change helps predict the energy requirement or release in chemical processes. It is crucial for designing industrial chemical reactions and determining reaction feasibility.

Internal Energy Change

Internal energy change (
ackslashmathrm{E}) reflects the total energy change within a system during a reaction. It accounts for all kinds of energy exchanges, not just heat, as in enthalpy. Enthalpy is related to internal energy through the equation: \[ 
ackslashDelta H = 
ackslashDelta E + 
ackslashDelta n_gRT \]Where 
ackslashDelta n_g is the change in moles of gas, 
ackslashmathrm{R} is the gas constant, and 
ackslashmathrm{T} is temperature in Kelvin.
In the provided solution, by rearranging and solving this equation, we find that 
ackslashDelta E is 17.8 kcal. Calculating internal energy change helps chemists understand the fundamental energy dynamics in reactions beyond just heat transfers.

Reaction Stoichiometry

Reaction stoichiometry is the balance of reactants and products in a chemical equation. It is essential for calculating how much of each substance is consumed or produced.
In this exercise, the balanced reaction equation is:\[ A(g) + 2 B(g) \rightarrow 2 C(g) + 3 D(g) \]From this, the total moles of gas on each side of the equation are needed to find 
ackslashDelta n_g. Here, 
ackslashDelta n_g is calculated as:

• Products: 2 moles of C(g) + 3 moles of D(g) = 5 moles
• Reactants: 1 mole of A(g) + 2 moles of B(g) = 3 moles
• So, 
  ackslashDelta n_g = 5 - 3 = 2

Proper stoichiometry ensures accurate calculations of mass and energy changes in chemical reactions.

Gas Constant

The gas constant (
ackslashmathrm{R}) is a fundamental value in thermodynamics. It relates energy and temperature, facilitating the conversion in various gas law equations. Here, the gas constant value is provided as 2.0 cal K mol
. It plays a critical role in translating the energy changes involved when gases react, as seen in the equation:\[ 
ackslashDelta H = 
ackslashDelta E + 
ackslashDelta n_gRT \]
In the context of the exercise, 
ackslashmathrm{R} is used to adjust for temperature when calculating the energy change between enthalpy and internal energy. The gas constant helps bridge the macroscopic and microscopic understandings of gases by integrating their thermal behaviors into calculations.