Question

A certain reaction has \(\Delta H^{\circ}=+23.7 \mathrm{kJ}\) and \(\Delta S^{\circ}=\) \(+52.4 \mathrm{J} / \mathrm{K}\) . (a) Is the reaction exothermic or endothermic? (b) Does the reaction lead to an increase or decrease in the randomness or disorder of the system?(c) Calculate \(\Delta G^{\circ}\) for the reaction at 298 \(\mathrm{K}\) . (d) Is the reaction spontaneous at 298 \(\mathrm{K}\) under standard conditions?

Short answer

(a) The reaction is endothermic because \(\Delta H^{\circ} = +23.7 \,\mathrm{kJ}\) is positive. (b) The reaction leads to an increase in the randomness or disorder of the system since \(\Delta S^{\circ} = +52.4 \,\mathrm{J/K}\) is positive. (c) The standard Gibbs free energy change \(\Delta G^{\circ} = 8,084.8\,\mathrm{J}\) at 298 K. (d) The reaction is non-spontaneous at 298 K under standard conditions because \(\Delta G^{\circ} = 8,084.8\,\mathrm{J}\) is positive.

Step-by-step solution

Part (a): Determine if the reaction is exothermic or endothermic

To determine if the reaction is exothermic or endothermic, we need to look at the sign of \(\Delta H^{\circ}\): - If \(\Delta H^{\circ} < 0\), the reaction is exothermic. - If \(\Delta H^{\circ} > 0\), the reaction is endothermic. We are given that \(\Delta H^{\circ} = +23.7 \,\mathrm{kJ}\). Since this value is positive, we can conclude that the reaction is endothermic.

Part (b): Determine if the reaction increases or decreases the randomness or disorder of the system

To determine if the reaction increases or decreases the system's randomness or disorder, we need to look at the sign of \(\Delta S^{\circ}\): - If \(\Delta S^{\circ} < 0\), the reaction results in a decrease in the randomness or disorder of the system. - If \(\Delta S^{\circ} > 0\), the reaction results in an increase in the randomness or disorder of the system. We are given that \(\Delta S^{\circ} = +52.4 \,\mathrm{J/K}\). Since this value is positive, we can conclude that the reaction leads to an increase in the randomness or disorder of the system.

Part (c): Calculate the standard Gibbs free energy change \(\Delta G^{\circ}\) for the reaction at 298 K

To calculate the standard Gibbs free energy change \(\Delta G^{\circ}\) for the reaction, we use the following equation: \[\Delta G^{\circ} = \Delta H^{\circ} - T\Delta S^{\circ}\] We are given that \(\Delta H^{\circ} = +23.7 \,\mathrm{kJ}\), \(\Delta S^{\circ} = +52.4 \,\mathrm{J/K}\), and \(T = 298 \,\mathrm{K}\). First, let's convert the value of \(\Delta H^{\circ}\) to J by multiplying it by 1000: \[\Delta H^{\circ} = 23.7 \,\mathrm{kJ} \cdot 1000 = 23,700 \,\mathrm{J}\] Now we can substitute the given values into the equation and calculate \(\Delta G^{\circ}\): \[\Delta G^{\circ} = 23,700\,\mathrm{J} - (298 \,\mathrm{K})(52.4\,\mathrm{J/K})\] \[\Delta G^{\circ} = 23,700\,\mathrm{J} - 15,615.2\,\mathrm{J}\] \[\Delta G^{\circ} = 8,084.8\,\mathrm{J}\]

Part (d): Determine if the reaction is spontaneous at 298 K under standard conditions

To determine if the reaction is spontaneous at 298 K under standard conditions, we need to look at the sign of \(\Delta G^{\circ}\): - If \(\Delta G^{\circ} < 0\), the reaction is spontaneous. - If \(\Delta G^{\circ} > 0\), the reaction is non-spontaneous. We have calculated that \(\Delta G^{\circ} = 8,084.8\,\mathrm{J}\). Since this value is positive, we can conclude that the reaction is non-spontaneous at 298 K under standard conditions.

Key concepts

Endothermic Reaction

An endothermic reaction is a process that absorbs energy from its surroundings, usually in the form of heat. This can be understood by examining the term itself, where 'endo-' means 'within' and 'thermic' refers to 'heat'. In such reactions, the total energy of the products is greater than that of the reactants, leading to a positive change in enthalpy (go{H}). This is indicated by a (+) sign in the value of H, as was the case in our exercise problem where go{H} = +23.7 kJ.
In our daily lives, endothermic reactions are quite common. For instance, when you use a cold pack from a first-aid kit, the chemical reaction that occurs inside the pack absorbs heat from your body, providing the cooling effect. Understanding that an endothermic reaction requires energy input to proceed helps us recognize that these reactions may not always occur spontaneously, as they might need an energy source to drive them forward.
It's crucial for students to recognize the energetic aspects of a reaction when analyzing its properties, as these have direct consequences on the reaction's spontaneity and feasibility under given conditions.

Entropy

Entropy, symbolized as S, is a thermodynamic quantity that represents the degree of randomness or disorder within a system. The second law of thermodynamics states that the entropy of an isolated system never decreases over time, and systems tend to evolve towards a state of maximum entropy.
When examining a chemical reaction, a positive change in entropy (S > 0) indicates that the products of the reaction are more disordered than the reactants. This was exemplified in our exercise, with S being +52.4 J/K, signifying an increase in disorder. This is analogous to a room becoming messier over time if no effort is made to tidy it up; the natural progression favors disorder.
In any process, the entropy of the universe (S of the surroundings plus the system) increases, which drives the direction of natural processes. This increase can be due to the dispersion of energy or matter. An understanding of entropy is crucial for students as it intimately connects with the concept of energy distribution within a chemical reaction and the possibility of the reaction occurring spontaneously.

Spontaneous Reaction

A spontaneous reaction is a process that occurs under a specific set of conditions without the need for continuous external energy. Spontaneity does not imply that a reaction occurs rapidly, only that it is thermodynamically favored to proceed. The determinant factor for spontaneity in a reaction is the change in Gibbs free energy, denoted as G. The condition for a spontaneous reaction is that G must be negative (G < 0).
G incorporates both enthalpy and entropy changes within a system, defined by the equation G = H - TS, where T is the temperature in Kelvin. From the given exercise, despite an increase in entropy, the reaction was determined to be non-spontaneous at 298 K since G was calculated to be positive.
It's essential for students to understand the concept of free energy since it provides a powerful insight into the direction of chemical reactions and their equilibrium states. The interplay between enthalpy, entropy, and temperature in determining spontaneity is a fundamental aspect of chemical thermodynamics.

Thermodynamics

Thermodynamics is the branch of physical science concerned with heat and its relation to other forms of energy and work. It dictates the principles that govern the transformation of energy within a system. The main laws of thermodynamics describe how energy moves and changes form, such as how it is conserved (the first law), increasingly spreads out (the second law), and cannot be completely converted into useful work (the third law).
Understanding thermodynamics is essential for any student of the physical sciences because it forms the foundation for various scientific and engineering principles, from mechanical engines to chemical reactions. It not only deals with the concepts of heat and work but also with the study of energy transfers, the efficiency of systems, and the spontaneity of processes. Remember that thermodynamics can predict the feasibility of reactions under different conditions and plays a key role in fields such as chemistry, physics, and even biology.