Problem 86 - A Complete Resource Book in Chemistry for JEE Main

Question

If the standard entropies of \(\mathrm{CH}_{4}(\mathrm{~g}), \mathrm{H}_{2} \mathrm{O}(\mathrm{g}), \mathrm{CO}_{2}(\mathrm{~g})\) and \(\mathrm{H}_{2}(\mathrm{~g})\) are \(186.2,188.2,197.6\) and \(130.6 \mathrm{JK}^{-1} \mathrm{~mol}^{-1}\) respectively, then the standard entropy change for the reaction \(\mathrm{CH}_{4}(\mathrm{~g})+\mathrm{H}_{2} \mathrm{O}(\mathrm{g}) \longrightarrow \mathrm{CO}_{2}(\mathrm{~g})+3 \mathrm{H}_{2}(\mathrm{~g})\) is (a) \(215 \mathrm{JK}^{-1} \mathrm{~mol}^{-1}\) (b) \(225 \mathrm{JK}^{-1} \mathrm{~mol}^{-1}\) (c) \(145 \mathrm{JK}^{-1} \mathrm{~mol}^{-1}\) (d) \(285 \mathrm{JK}^{-1} \mathrm{~mol}^{-1}\)

Step-by-Step Solution

Answer

The standard entropy change for the reaction is 215 JK\(^{-1}\) mol\(^{-1}\).

Step 1: Write the Reaction Equation

The given reaction is \( \mathrm{CH}_{4}(\mathrm{g}) + \mathrm{H}_{2}O(\mathrm{g}) \longrightarrow \mathrm{CO}_{2}(\mathrm{g}) + 3 \mathrm{H}_{2}(\mathrm{g}) \). Thus, we identify the reactants and products.

Step 2: List Standard Entropies

From the problem statement, the standard molar entropies are provided as follows: \( S^\circ_{\mathrm{CH}_4(g)} = 186.2 \, \mathrm{JK}^{-1} \mathrm{mol}^{-1} \), \( S^\circ_{\mathrm{H}_2O(g)} = 188.2 \, \mathrm{JK}^{-1} \mathrm{mol}^{-1} \), \( S^\circ_{\mathrm{CO}_2(g)} = 197.6 \, \mathrm{JK}^{-1} \mathrm{mol}^{-1} \), \( S^\circ_{\mathrm{H}_2(g)} = 130.6 \, \mathrm{JK}^{-1} \mathrm{mol}^{-1} \).

Step 3: Calculate Total Entropy of Products

For the products \( \mathrm{CO}_2(\mathrm{g}) \) and \( 3\,\mathrm{H}_2(\mathrm{g}) \), calculate the total entropy: \( S^\circ_{products} = S^\circ_{\mathrm{CO}_2(g)} + 3 \times S^\circ_{\mathrm{H}_2(g)} = 197.6 + 3 \times 130.6 = 589.4 \, \mathrm{JK}^{-1} \mathrm{mol}^{-1} \).

Step 4: Calculate Total Entropy of Reactants

Sum up the standard entropies of the reactants, \( \mathrm{CH}_4(\mathrm{g}) \) and \( \mathrm{H}_2O(\mathrm{g}) \): \( S^\circ_{reactants} = S^\circ_{\mathrm{CH}_4(g)} + S^\circ_{\mathrm{H}_2O(g)} = 186.2 + 188.2 = 374.4 \, \mathrm{JK}^{-1} \mathrm{mol}^{-1} \).

Step 5: Calculate Standard Entropy Change

The standard entropy change \( \Delta S^\circ \) for the reaction is determined by subtracting the total entropy of reactants from the total entropy of products: \( \Delta S^\circ = S^\circ_{products} - S^\circ_{reactants} = 589.4 - 374.4 = 215 \, \mathrm{JK}^{-1} \mathrm{mol}^{-1} \).

Key Concepts

Standard Entropy

In thermodynamics, standard entropy is an important concept used to describe the degree of disorder or randomness in a system. Each chemical substance has a specific standard molar entropy, which is the entropy content of one mole of a substance under standard conditions.
Standard conditions typically refer to a pressure of one atmosphere and a specified temperature, often 298.15 K (25°C). The unit for standard entropy is usually expressed in joules per kelvin per mole ( \(\mathrm{JK}^{-1} \mathrm{mol}^{-1}\) ).
Knowing the standard entropy values of substances helps chemists predict the feasibility and spontaneity of chemical reactions. For instance, in a reaction, if the total entropy of the products is greater than the total entropy of the reactants, then the process tends to lead to an increased disorder, which usually means a spontaneous reaction under constant temperature and pressure.

Chemical Reactions

Chemical reactions involve the transformation of reactants into products, and during this process, atoms are rearranged. This transformation can lead to changes in the energy and entropy of the system. Understanding these changes is crucial for predicting the direction and extent of reactions.
For our reaction \( \mathrm{CH}_4(\mathrm{g}) + \mathrm{H}_2O(\mathrm{g}) \rightarrow \mathrm{CO}_2(\mathrm{g}) + 3 \mathrm{H}_2(\mathrm{g})\), reactants such as methane and water vapor react to form carbon dioxide and hydrogen gas.

• Methane (\( \mathrm{CH}_4\)) is a major reactant often found in combustion reactions.
• Water vapor (\( \mathrm{H}_2O\)) contributes to environmental reactions.
• Carbon dioxide (\( \mathrm{CO}_2\)) is a common product in many combustion reactions.
• Hydrogen gas (\(\mathrm{H}_2\)) is used in hydrogenation processes and as a clean energy source.

Analyzing a reaction involves evaluating the energetic and entropic aspects to determine how likely a reaction is to proceed under given conditions. The balance between energy and entropy changes provides insight into the reaction's spontaneity and direction.

Entropy Change

Entropy change (\( \Delta S \)) is a measure of the change in disorder as a reaction proceeds. It can be considered as the difference between the sum of the standard entropies of the products and the sum of the standard entropies of the reactants.
For the given reaction \( \mathrm{CH}_4(\mathrm{g}) + \mathrm{H}_2O(\mathrm{g}) \rightarrow \mathrm{CO}_2(\mathrm{g}) + 3 \mathrm{H}_2(\mathrm{g})\), the standard entropy change (\( \Delta S^\circ \)) can be calculated to predict the reaction's spontaneity using its standard entropies.
The calculation of \( \Delta S^\circ \) involves:

• Adding up standard entropies of products such as \( \mathrm{CO}_2 \) and \( 3\mathrm{H}_2 \):
• Subtracting standard entropies of reactants like \( \mathrm{CH}_4 \) and \( \mathrm{H}_2O \).

In this particular reaction, the calculated \( \Delta S^\circ \) is \( 215 \, \mathrm{ JK}^{-1} \mathrm{mol}^{-1} \), suggesting that the entropy of the system increases, indicating a likely spontaneous process. Understanding these changes is critical for predictions about reaction feasibility and equilibrium.