Problem 52 - A Complete Resource Book in Chemistry for JEE Main

Question

The enthalpy and entropy change for the reaction, \(\mathrm{Br}_{2}(\mathrm{l})+\mathrm{Cl}_{2}(\mathrm{~g})\) \(2 \mathrm{BrCl}(\mathrm{g})\) are \(30 \mathrm{~kJ} \mathrm{~mol}^{-1}\) and \(105 \mathrm{~J} \mathrm{~mol}^{-1}\) respectively. The temperature at which the reaction will be in equilibrium is (a) 450 (b) 300 (c) \(285.7\) (d) 273

Step-by-Step Solution

Answer

The equilibrium temperature is approximately 285.7 K.

Step 1: Understand the relationship between Gibbs free energy and equilibrium

To find the temperature at which the reaction is at equilibrium, we use the Gibbs free energy change formula: \( \Delta G = \Delta H - T \Delta S \). At equilibrium, \( \Delta G = 0 \). Therefore, the equation becomes \( 0 = \Delta H - T \Delta S \). Rearranging gives \( T = \frac{\Delta H}{\Delta S} \). This relationship will allow us to solve for the temperature at equilibrium.

Step 2: Convert entropy units if necessary

The change in entropy (\( \Delta S \)) is given in \( \text{J mol}^{-1} \), but we often work in \( \text{kJ mol}^{-1} \) in thermodynamics problems. Thus, we convert \( \Delta S = 105 \text{ J mol}^{-1} \) to \( \text{kJ mol}^{-1} \) by dividing by \( 1000 \). Therefore, \( \Delta S = 0.105 \text{ kJ mol}^{-1} \).

Step 3: Substitute values into the equilibrium formula

Now substitute the given values into the rearranged formula from Step 1: \( T = \frac{\Delta H}{\Delta S} \). With \( \Delta H = 30 \text{ kJ mol}^{-1} \) and \( \Delta S = 0.105 \text{ kJ mol}^{-1} \), substitute them to get \( T = \frac{30}{0.105} \).

Step 4: Calculate the temperature

Perform the division: \( \frac{30}{0.105} \approx 285.71 \text{ K} \). This result gives us the temperature at which the reaction is at equilibrium.

Key Concepts

Enthalpy

Enthalpy, symbolized as \( \Delta H \), is a fundamental concept in thermodynamics representing the total heat content of a system. It reflects the amount of heat absorbed or released during a chemical reaction at constant pressure. This makes it extremely valuable in predicting the heat flow between a system and its surroundings. When a reaction releases heat, it is exothermic, and when it absorbs heat, it is endothermic. The enthalpy change is usually expressed in kilojoules per mole (kJ/mol), helping us quantify how much energy is released or absorbed.

Understanding enthalpy is essential for determining energy requirements and efficiencies in chemical processes. It's crucial in industries where energy management is a priority, such as manufacturing and pharmaceuticals.

Entropy

Entropy, denoted as \( \Delta S \), measures the degree of disorder or randomness in a system. It's a key aspect that helps predict the direction of processes and the feasibility of reactions. In thermodynamics, we often relate it to energy dispersal; processes that increase entropy tend to happen naturally. In our exercise, entropy is given in joules per mole (J/mol) but is often converted to kilojoules per mole for consistency in calculations.

• Higher entropy: More disorder and randomness
• Lower entropy: More order and less randomness

By increasing entropy, systems achieve greater energic freedom, thus driving natural processes forward. This concept plays a crucial role in determining equilibrium states and understanding the changes during chemical reactions.

Equilibrium Temperature

Equilibrium temperature refers to the temperature at which a reaction's Gibbs free energy change \( \Delta G \) becomes zero, indicating no net change in the composition of reactants and products. This is the balancing act between enthalpy and entropy. The formula \( \Delta G = \Delta H - T \Delta S \) at equilibrium rearranges to \( T = \frac{\Delta H}{\Delta S} \), simplifying the calculation of this temperature.

The concept is fundamental in chemical thermodynamics for understanding the conditions necessary for a reaction to spontaneously maintain equilibrium without external energy input. Achieving this temperature is vital for reactions requiring precise conditions, such as biochemical processes and industrial syntheses.

Thermodynamics

Thermodynamics is the broad study of energy transformations, encompassing concepts like enthalpy and entropy. It fundamentally examines how energy changes between forms, particularly within chemical reactions. This field provides the framework for understanding reaction spontaneity through the Gibbs free energy equation, \( \Delta G = \Delta H - T \Delta S \).

• First Law of Thermodynamics: Energy cannot be created or destroyed, only transformed.
• Second Law of Thermodynamics: Total entropy in an isolated system will never decrease over time.

These laws guide the calculation and prediction of possible chemical processes and energy efficiency. Thermodynamics is not just theory; it influences practical applications like engine design, environmental control, and industrial processes.