Question

A particular reaction is spontaneous at $450\mathrm{K}$. The enthalpy change for the reaction is $+34.5\mathrm{kJ}$. What can you conclude about the sign and magnitude of $\mathrm{\Delta }S$ for the reaction?

Short answer

We can conclude that the entropy change (ΔS) for the reaction is positive and its magnitude is greater than 76.67 J/K.

Step-by-step solution

Write down the known variables and equation

We are given: - Reaction is spontaneous: ΔG < 0 - Temperature: T = 450 K - Enthalpy change: ΔH = +34.5 kJ We will be using the Gibbs Free Energy equation: $\Delta G=\Delta H-T\Delta S$

Plug in the known variables into the equation

We know that ΔG < 0, ΔH = +34.5 kJ, and T = 450 K, so the equation becomes: $0>\Delta H-T\Delta S$

Solve for ΔS

Now, we want to isolate ΔS in the equation: $0>\Delta H-T\Delta S$ $T\Delta S>\Delta H$ $\Delta S>{\displaystyle \frac{\Delta H}{T}}$ Now, substitute the given values of ΔH and T: $\Delta S>{\displaystyle \frac{+34.5\mathrm{kJ}}{450\mathrm{K}}}$ $\Delta S>{\displaystyle \frac{+34.5\times {10}^3\mathrm{J}}{450\mathrm{K}}}$ (converting kJ to J) $\Delta S>76.67\mathrm{J}/\mathrm{K}$

Conclusion:

Since ΔS is greater than 76.67 J/K, we can conclude that the entropy change (ΔS) for the reaction is positive and its magnitude is greater than 76.67 J/K.

Key concepts

Spontaneous Reactions

In chemistry, a reaction is deemed **spontaneous** if it occurs without an external input of energy upon reaching a certain state. Spontaneity is determined by the Gibbs Free Energy change ($\Delta G$ for the reaction. A reaction is spontaneous when $\Delta G<0$. This means the process releases free energy and can proceed on its own.
Understanding spontaneity is crucial for predicting whether chemical reactions will occur under specific conditions. For instance, if you're looking at a reaction at a given temperature and see that $\Delta G$ is less than zero, that means the reaction is spontaneous under those conditions. This prediction is based on the interplay of enthalpy change ($\Delta H$) and entropy change ($\Delta S$). To summarize, remember:

• Spontaneous means $\Delta G<0$.
• Spontaneity indicates a process can occur without needing more energy.
• Depends on $\Delta H$, $\Delta S$, and the temperature.

Enthalpy Change

**Enthalpy change**, denoted as $\Delta H$, reflects the total heat content variation within a system during a reaction. It can be either positive or negative, impacting the spontaneity of the reaction. A positive $\Delta H$ indicates that the reaction absorbs heat (endothermic), while a negative $\Delta H$ signifies heat release (exothermic).
In the given problem, the reaction had an $\Delta H$ of +34.5 kJ, meaning it absorbs heat. This alone suggests the reaction is endothermic. However, for it still to be spontaneous (as it is given as so), the increase in entropy must compensate for the energy absorbed leading to a negative $\Delta G$.
It is crucial to analyze how both $\Delta H$ and temperature influence whether a reaction is favorable or not. In brief:

• $\Delta H>0$: Endothermic (absorbs heat).
• $\Delta H<0$: Exothermic (releases heat).
• Spontaneity relies on balancing $\Delta H$ and $T\Delta S$.

Entropy Change

Entropy change ($\Delta S$) is a measure of the disorder or randomness in a system. It can influence whether a reaction will occur spontaneously at a particular temperature. Generally, an increase in entropy ($\Delta S>0$) means that the disorder of the system is increasing, favoring spontaneity in a broader range of temperatures.
In this problem, to find out how the entropy change allows the reaction to be spontaneous, we used the formula:$$\Delta G=\Delta H-T\Delta S$$Here, we know $\Delta G<0$, and having $\Delta H=+34.5kJ$ implies a high $\Delta S$ is required to maintain spontaneity since the reaction is endothermic. After calculations, we found that:$$\Delta S>76.67J/K$$This positive entropy change helps offset the positive enthalpy, ensuring the overall $\Delta G$ remains negative under the given conditions, highlighting how crucial entropy is in the spontaneity equation. In summary:

• $\Delta S>0$: Increases disorder, can aid in spontaneous reactions.
• $\Delta S$ significantly impacts the Gibbs Free Energy.
• Spontaneity is often the result of a balance between $\Delta H$ and $\Delta S$.