Problem 85 - A Complete Resource Book in Chemistry for JEE Main

Question

The standard entropies of \(\mathrm{CO}_{2}(\mathrm{~g}), \mathrm{C}(\mathrm{s})\) and \(\mathrm{O}_{2}(\mathrm{~g})\) are \(213.5,5.74\) and \(205 \mathrm{JK}^{-1}\) respectively. The standard entropy of the formation of \(\mathrm{CO}_{2}(\mathrm{~g})\) is (a) \(1.16 \mathrm{~J} \mathrm{~K}^{-1}\) (b) \(2.76 \mathrm{~J} \mathrm{~K}^{-1}\) (c) \(1.86 \mathrm{~J} \mathrm{~K}^{-1}\) (d) \(2.12 \mathrm{~J} \mathrm{~K}^{-1}\)

Step-by-Step Solution

Answer

The standard entropy change is 2.76 J K^{-1}, corresponding to option (b).

Step 1: Understand the Reaction

The reaction for the formation of carbon dioxide from its elements is: \[ \mathrm{C} (\mathrm{s}) + \mathrm{O}_2 (\mathrm{g}) \rightarrow \mathrm{CO}_2 (\mathrm{g}) \] We need to calculate the standard entropy change \(\Delta S^\circ\) for this reaction.

Step 2: Identify the Formula for Entropy Change

The standard entropy change for a reaction is given by the formula: \[ \Delta S^\circ = S^\circ_{\text{products}} - S^\circ_{\text{reactants}} \] where \( S^\circ \) represents the standard entropy of the substance.

Step 3: Plug in the Values for Entropy

For the reaction, the standard entropies are given as: \( S^\circ_{\mathrm{CO}_2} = 213.5 \, \mathrm{J} \mathrm{K}^{-1} \), \( S^\circ_{\mathrm{C}} = 5.74 \, \mathrm{J} \mathrm{K}^{-1} \), and \( S^\circ_{\mathrm{O}_2} = 205 \, \mathrm{J} \mathrm{K}^{-1} \). Substitute these into the formula: \[ \Delta S^\circ = 213.5 - (5.74 + 205) \]

Step 4: Solve for Entropy Change

Calculate the values: \( 5.74 + 205 = 210.74 \). Then compute the entropy change: \[ \Delta S^\circ = 213.5 - 210.74 = 2.76 \, \mathrm{J} \mathrm{K}^{-1} \]

Step 5: Choose the Correct Option

Compare the calculated \( \Delta S^\circ = 2.76 \, \mathrm{J} \mathrm{K}^{-1} \) with the given options to find that (b) matches our answer.

Key Concepts

Entropy Change Formula

When chemical reactions take place, entropies of the involved substances change. Entropy, symbolized as \( S \), is a measure of energy distribution in a system. In thermodynamics, figuring out how entropy changes during reactions helps maintain energy balance and predict reaction feasibility. To calculate the standard entropy change, \( \Delta S^\circ \), for a given reaction, we use the formula:\[ \Delta S^\circ = S^\circ_{\text{products}} - S^\circ_{\text{reactants}} \]Here, \( S^\circ \) is the standard entropy of each substance, provided under standard conditions (1 atm, 298 K). This formula reflects the change from reactants to products. For balanced chemical equations, ensure all stoichiometric coefficients are correctly applied when using this formula. Comparing the sum of products' entropies with the reactants' gives \( \Delta S^\circ \), indicating whether disorder increases.

Standard Entropies

Standard entropies, depicted as \( S^\circ \), are thermodynamic quantities that measure the amount of disorder or randomness in a substance under standard conditions. Unlike energy, which can only increase, entropy measures a system's ability to spread and make other entities accessible. Carbon dioxide \( (\mathrm{CO}_2) \), carbon \( (\mathrm{C}) \), and oxygen \( (\mathrm{O}_2) \) have standard entropies expressed in \( \mathrm{J} \mathrm{K}^{-1} \). For example:

• \( \mathrm{CO}_2: 213.5 \, \mathrm{J} \mathrm{K}^{-1} \)
• \( \mathrm{C}: 5.74 \, \mathrm{J} \mathrm{K}^{-1} \)
• \( \mathrm{O}_2: 205 \, \mathrm{J} \mathrm{K}^{-1} \)

These values provide a baseline for predicting how systems will evolve thermodynamically. The higher the entropy value, the more dispersed its energy distribution.

Carbon Dioxide Formation

The reaction involving the formation of carbon dioxide \( \mathrm{CO}_2 \) from its constituent elements is simple yet significant in thermodynamics. It demonstrates entropy changes when solid carbon \( \mathrm{C} \) and gaseous oxygen \( \mathrm{O}_2 \) combine:\[ \mathrm{C} (\mathrm{s}) + \mathrm{O}_2 (\mathrm{g}) \rightarrow \mathrm{CO}_2 (\mathrm{g}) \]This reaction highlights two main points:

• Conversion from ordered solid (in carbon) and diatomic molecules (oxygen) to a single gaseous product \( \mathrm{CO}_2 \).
• Understanding \( \mathrm{CO}_2 \)'s formation assists in energy system evaluations due to its widespread occurrence and impact on climate.

When calculating standard entropy changes, recognize this reaction provides insight into a complete energy state shift from separated to a united form.

Thermodynamics

Thermodynamics explores how energy and matter interact under various conditions. A core principle involves analyzing how systems transfer energy into work and heat. In this subject, entropy establishes the direction and spontaneity of reactions. Key principles include:

• The First Law (energy conservation), which states energy cannot be created or destroyed.
• The Second Law, introducing entropy, states that every energy transfer increases the total entropy of the universe.

These laws suggest why reactions occur, how energy is distributed, and the formation of chemical compounds like \( \mathrm{CO}_2 \). For instance, forming \( \mathrm{CO}_2 \) from \( \mathrm{C} \) and \( \mathrm{O}_2 \) aligns with increasing entropy, showcasing a transition to a more stable state. Understanding these fundamentals helps predict how chemical systems react to external changes.