Question

The standard enthalpy of decomposition of \(\mathrm{N}_{2} \mathrm{O}_{4}\) to \(\mathrm{NO}_{2}\) is \(58.04 \mathrm{~kJ}\) and standard entropy of this reaction is \(176.7 \mathrm{~J} \mathrm{~K}^{-1}\). The standard free energy change for this reaction at \(25^{\circ} \mathrm{C}\), is: (a) \(5.39 \mathrm{~kJ}\) (b) \(-5.39 \mathrm{~kJ}\) (c) \(539 \mathrm{~kJ}\) (d) \(53.9 \mathrm{~kJ}\)

Short answer

The standard free energy change is \\(5.39 \, \text{kJ.}\\)

Step-by-step solution

Understand the Gibbs Free Energy Equation

The Gibbs free energy change (\( \Delta G^{\circ} \)) for a reaction can be calculated using the equation: \[ \Delta G^{\circ} = \Delta H^{\circ} - T\Delta S^{\circ} \]where \( \Delta H^{\circ} \) is the standard enthalpy change, \( \Delta S^{\circ} \) is the standard entropy change, and \( T \) is the temperature in Kelvin.

Convert Temperature to Kelvin

The temperature given is \(25^{\circ} \mathrm{C}\). To use the equation for \( \Delta G^{\circ} \), temperature must be in Kelvin. Convert Celsius to Kelvin using the formula: \[ T(K) = T(^{\circ}\mathrm{C}) + 273.15 \]For \(25^{\circ} \mathrm{C}\), the conversion is: \[ 25 + 273.15 = 298.15 \text{ K} \]

Key concepts

Standard Enthalpy Change

The standard enthalpy change, denoted as \( \Delta H^{\circ} \), is an important concept in thermodynamics. It represents the heat absorbed or released by a reaction at constant pressure, under standard conditions, which are typically 1 atmosphere of pressure and a temperature of 298 K (25°C).

• Standard conditions ensure consistency when comparing enthalpy changes between different reactions.
• An endothermic reaction (absorbing heat) will have a positive \( \Delta H^{\circ} \), whereas an exothermic reaction (releasing heat) will have a negative \( \Delta H^{\circ} \).

In the exercise provided, the standard enthalpy change is given as \(58.04 \text{ kJ} \). This indicates that the decomposition of \( \mathrm{N}_{2} \mathrm{O}_{4} \) to \( \mathrm{NO}_{2} \) is an endothermic process, meaning it requires heat to proceed.

Standard Entropy Change

Standard entropy change, represented as \( \Delta S^{\circ} \), measures the change in disorder or randomness of a system. Entropy is often associated with the second law of thermodynamics, which states that processes naturally progress towards increased disorder.

• A positive \( \Delta S^{\circ} \) suggests an increase in disorder, while a negative \( \Delta S^{\circ} \) suggests a decrease.
• Entropy changes in chemical reactions can be influenced by factors such as the physical state of reactants and products and their mole ratios.

For the given reaction, the standard entropy change is \(176.7 \mathrm{~J} \mathrm{~K}^{-1}\). A positive value, like this one, indicates that the formation of \( \mathrm{NO}_{2} \) from \( \mathrm{N}_{2} \mathrm{O}_{4} \) results in a more disordered system.

Temperature Conversion to Kelvin

Temperature conversion to Kelvin is a crucial step in thermodynamic calculations since most equations, like the Gibbs free energy equation, require temperature to be expressed in Kelvin.

• The Kelvin scale is absolute, starting at absolute zero, which is the theoretical point where all molecular motion ceases.
• To convert from Celsius to Kelvin, simply add the constant 273.15. The formula is: \( T(K) = T(^{\circ}\mathrm{C}) + 273.15 \).

In the problem, we are given \(25^{\circ} \mathrm{C}\). By converting to Kelvin, we calculate \(25 + 273.15 = 298.15 \text{ K}\), a fundamental step needed to apply the Gibbs free energy equation accurately.