Question

A certain reaction is nonspontaneous at $-{25}^{\circ }\mathrm{C}$. The entropy change for the reaction is $95\mathrm{J}/\mathrm{K}$. What can you conclude about the sign and magnitude of $\mathrm{\Delta }H?$

Short answer

The sign of the enthalpy change (∆H) for the reaction is positive, and the magnitude of ∆H must be greater than 23574.25 J for the reaction to be nonspontaneous at -25°C (248.15 K).

Step-by-step solution

Convert temperature to Kelvin

First, if given temperature in Celsius, convert it into Kelvin. To convert -25°C to Kelvin, use the equation: $$T(K)=T(°C)+273.15$$ $$T(K)=-25+273.15$$ $$T(K)=248.15$$ So, the temperature in Kelvin is 248.15 K.

Analyze the sign of ∆H using ∆G and ∆S

Since the reaction is nonspontaneous, we know that ∆G > 0. We also know that ∆S is positive from the given value (95 J/K). Now, recall the Gibbs free energy equation: $$∆G=∆H–T∆S$$ To maintain ∆G > 0, given that T∆S is positive, ∆H must be positive as well, so: ∆G > 0 → ∆H > T∆S

Calculate a lower limit for the magnitude of ∆H

We know that ∆H > T∆S, so let's find the lower limit for the magnitude of ∆H by calculating T∆S. $$T∆S=(248.15\mathrm{K})(95\mathrm{J}/\mathrm{K})$$ $$T∆S=23574.25\mathrm{J}$$ Since ∆H > T∆S, we can conclude that the magnitude of ∆H must be greater than 23574.25 J. In summary, the sign of the enthalpy change (∆H) for the reaction is positive, and the magnitude of ∆H must be greater than 23574.25 J for the reaction to be nonspontaneous at -25°C (248.15 K).

Key concepts

Nonspontaneous Reaction

In chemistry, understanding when a reaction is nonspontaneous is crucial. A nonspontaneous reaction is one that does not occur on its own without the addition of external energy. This type of reaction is characterized by a positive Gibbs free energy change, denoted as $\mathrm{\Delta }G>0$. This implies that the process is energetically unfavorable and requires some form of energy input to proceed.

For instance, in the exercise provided, we determine that the reaction at $-{25}^{\circ }\mathrm{C}$ is nonspontaneous. This conclusion stems from the fact that the calculation of the Gibbs free energy, $\mathrm{\Delta }G$, where $\mathrm{\Delta }G=\mathrm{\Delta }H-T\mathrm{\Delta }S$, results in a positive value. Therefore, without any external assistance, the chemical reaction will not move forward in the desired direction.

Some key features of nonspontaneous reactions include:

• Positive Gibbs Free Energy: As discussed, when the Gibbs free energy is more than zero, the reaction isn't spontaneous.
• Endothermic Requirement: Often requires an input of energy, such as heat or light, for the reaction to proceed.
• Equilibrium Position: The equilibrium of the reaction lies towards the reactants, indicating that the products are not favored under standard conditions.

Understanding these points will make it easier to predict whether a reaction will require external energy to occur.

Entropy Change

Entropy, denoted as $\mathrm{\Delta }S$, is a fundamental concept representing the disorder or randomness of a system. When we talk about entropy change in a reaction, we are referring to how the disorder in a system increases or decreases as a result of the chemical process.

In the context of our original problem, the entropy change is given as positive, $\mathrm{\Delta }S=95\mathrm{J}/\mathrm{K}$. This indicates that the disorder in the system increases as the reaction occurs. That is, the products of the reaction are more disordered compared to the reactants.

Entropy change plays a crucial role in predicting a reaction's spontaneity:

• Positive Entropy Change: An increase in entropy (positive $\mathrm{\Delta }S$) favors spontaneity, as systems naturally evolve towards greater disorder.
• Gibbs Free Energy Relation: The contribution of entropy to Gibbs free energy is negative when $T\mathrm{\Delta }S$ is positive. This can offset a positive enthalpy to create a spontaneous condition. However, if $\mathrm{\Delta }G$ remains positive, the reaction persists as nonspontaneous.

Despite positive entropy change, a nonspontaneous reaction indicates the other factors like enthalpy and temperature heavily influence the overall energy balance.

Enthalpy Change

Enthalpy, symbolized as $\mathrm{\Delta }H$, represents the total heat content of a system. It is essentially the energy needed to create a system out of nothing and influences whether a reaction is exothermic (releases heat) or endothermic (absorbs heat).

In our given scenario, the enthalpy change is positive. This indicates an endothermic reaction where heat is absorbed from the surroundings. This energy absorption is needed to drive the reaction forward, which aligns with the nature of nonspontaneous processes.

In comprehending enthalpy change, consider these points:

• Positive Enthalpy Change: Means energy is absorbed, and the reaction is endothermic.
• Relation to Temperature and Entropy: A positive $\mathrm{\Delta }H$ necessitates a larger positive $T\mathrm{\Delta }S$ component to make the reaction spontaneous. When $\mathrm{\Delta }H>T\mathrm{\Delta }S$, the reaction remains nonspontaneous.
• Magnitude of $\mathrm{\Delta }H$: In our exercise, $\mathrm{\Delta }H$ was found to be greater than $23,574.25$ J. This significant energy requirement is a clear indicator of the reaction’s need for external energy input.

The interplay of enthalpy with entropy and temperature is a fundamental facet of determining spontaneity and reaction feasibility.