Problem 176 - A Complete Resource Book in Chemistry for JEE Main

Question

For complete combustion of ethanol, \(\mathrm{C}_{2} \mathrm{H}_{5} \mathrm{OH} \ell+1\) \(3 \mathrm{O}_{2}(\mathrm{~g}) \rightarrow 2 \mathrm{CO}_{2}(\mathrm{~g})+3 \mathrm{H}_{2} \mathrm{O} \ell\) the amount of heat pro- duced as measured in bomb calorimeter, is \(1364.47 \mathrm{~kJ}\) \(\mathrm{mol}^{-1}\) at \(25^{\circ} \mathrm{C}\). Assuming ideality the Enthalpy of combustion, \(\Delta_{\mathrm{c}} \mathrm{H}\) for the reaction will be: [2014] \(\left(\mathrm{R}=8.314 \mathrm{~kJ} \mathrm{~mol}^{-1}\right)\) (a) \(-1460.50 \mathrm{kj} \mathrm{mol}^{-1}\) (b) \(-1350.50 \mathrm{~kJ} \mathrm{~mol}^{-1}\) (c) \(-1366.95 \mathrm{~kJ} \mathrm{~mol}^{-1}\) (d) \(-1361.95 \mathrm{~kJ} \mathrm{~mol}^{-1}\)

Step-by-Step Solution

Answer

The enthalpy of combustion is \(-1366.95 \text{ kJ/mol}\). Choose option (c).

Step 1: Understand the Reaction and Data Provided

We are analyzing the combustion of ethanol, represented by the balanced chemical equation: \( \text{C}_2\text{H}_5\text{OH}_{(\ell)} + 3 \text{O}_2_{(g)} \rightarrow 2\text{CO}_2_{(g)} + 3\text{H}_2\text{O}_{(\ell)} \). The heat measured by the bomb calorimeter for this reaction is \( 1364.47 \, \text{kJ/mol} \). Our goal is to determine the enthalpy of combustion, \( \Delta_c H \), considering that the value given is measured at constant volume.

Step 2: Recognize the Conditions

The heat provided by the bomb calorimeter, \( 1364.47 \, \text{kJ/mol} \), is the energy change at constant volume, which is equal to the change in internal energy, \( \Delta U \). We need to convert this \( \Delta U \) to the enthalpy change \( \Delta H \), using the equation \( \Delta H = \Delta U + \Delta n_gRT \). Here, the temperature \( T \) is \( 25^{\circ}C \) or \( 298 K \), and \( R = 8.314 \, \text{J} \, \text{mol}^{-1} \, \text{K}^{-1} \).

Step 3: Calculating \( \Delta n_g \)

Determine the change in moles of gas, \( \Delta n_g \), during the reaction. The reactant side has 3 moles of gas (from \( \text{O}_2 \)) and the product side also has 2 moles of \( \text{CO}_2 \). Thus, \( \Delta n_g = 2 - 3 = -1 \).

Step 4: Calculate the \( \Delta n_gRT \) Term

Convert \( R \) into \( \text{kJ} \) by dividing by 1000: \( R = 0.008314 \, \text{kJ} \, \text{mol}^{-1} \, \text{K}^{-1} \). Then calculate \( \Delta n_gRT = -1 \times 0.008314 \, \text{kJ} \, \text{mol}^{-1} \, \text{K}^{-1} \times 298 \text{K} = -2.477572 \text{kJ/mol} \).

Step 5: Determine \( \Delta_c H \)

Substitute \( \Delta U = -1364.47 \text{kJ/mol} \) and \( \Delta n_gRT = -2.477572 \text{kJ/mol} \) into the equation \( \Delta_c H = \Delta U + \Delta n_gRT \). Thus, \( \Delta_c H = -1364.47 \text{kJ/mol} - 2.477572 \text{kJ/mol} = -1366.947572 \text{kJ/mol} \).

Step 6: Select the Closest Option

Round \( -1366.947572 \text{kJ/mol} \) to a reasonable significant figure. The closest answer choice is \( -1366.95 \text{kJ/mol} \), option (c).

Key Concepts

Bomb Calorimeter

A bomb calorimeter is a device used to measure the heat of combustion of a chemical reaction. It is particularly significant in determining energies like enthalpy of combustion, which is essential in chemical thermodynamics. The device consists of a sealed container called a "bomb," where the reaction takes place. The bomb is immersed in a water bath to absorb heat released from the reaction.

How it Works:

• The sample is placed in the bomb chamber and oxygen is added to ensure complete combustion.
• The bomb is then sealed and placed in a water bath with known volume and temperature.
• The reaction is ignited electronically, causing the temperature of the water bath to increase.
• The change in water temperature is used to calculate the heat released by the reaction, using the specific heat capacity of water.

The heat measured by the bomb calorimeter represents the change in internal energy at constant volume, denoted as \( \Delta U \). To find the enthalpy change \( \Delta H \), we must convert this value using the equation \( \Delta H = \Delta U + \Delta n_gRT \), where \( \Delta n_g \) is the change in moles of gas, \( R \) is the gas constant, and \( T \) is the temperature in Kelvin.

Ideal Gas Law

The Ideal Gas Law is a fundamental equation in chemical thermodynamics and plays a critical role in calculating the enthalpy of combustion, specifically when determining changes in gas quantities. The Ideal Gas Law is given by the equation \( PV = nRT \), where:

• \( P \) is the pressure of the gas.
• \( V \) is the volume of the gas.
• \( n \) is the amount of substance in moles.
• \( R \) is the ideal gas constant \( 8.314 \, \text{J/mol} \cdot \text{K} \).
• \( T \) is the absolute temperature in Kelvin.

In the context of a bomb calorimeter experiment, the Ideal Gas Law helps us understand the behavior of gaseous reactants and products. For instance, when we calculate \( \Delta n_g \), the change in moles of gas during the reaction, it involves comparing the moles of gas before and after the combustion reaction, crucial for adjusting the internal energy \( \Delta U \) to find \( \Delta H \). This ensures a comprehensive understanding of heat exchanges under constant volume conditions.

Chemical Thermodynamics

Chemical thermodynamics studies heat and energy changes during chemical reactions. Understanding enthalpy of combustion, measured through processes like bomb calorimetry, involves applying principles of thermodynamics such as the First Law, which relates to the conservation of energy.

Thermodynamic Concepts:

• **Enthalpy (\( \Delta H \)):** Represents heat change at constant pressure, differing from internal energy (\( \Delta U \)).
• **Internal Energy (\( \Delta U \)):** At constant volume, as measured by bomb calorimeter, it refers to energy changes within the system, excluding work done.
• **Gas Expansion Work:** For reactions involving gases, the work done by expanding or contracting gases is significant when relating \( \Delta U \) to \( \Delta H \).

In a thermodynamic analysis of a combustion reaction like that of ethanol, it's essential to apply these concepts to calculate \( \Delta H \). The resulting figure represents the enthalpy change when one mole of ethanol combusts completely under specified conditions, crucial for understanding energy requirements and outputs in chemical processes.