Question

With its components in their standard states, a certain reaction is spontaneous only at high \(T .\) What do you know about the signs of \(\Delta H^{\circ}\) and \(\Delta S^{\circ}\) ? Describe a process for which this is true.

Short answer

For the reaction to be spontaneous at high temperatures, \(\triangle H > 0 \) and \(\triangle S > 0 \). A process fitting this is the melting of ice.

Step-by-step solution

Understand Spontaneity Criteria

For a reaction to be spontaneous at high temperatures, we need to use the Gibbs Free Energy equation: \[ \triangle G = \triangle H - T \triangle S \]A negative \(\triangle G\) indicates a spontaneous reaction.

Analyze High Temperature Conditions

At high temperatures, the term \( -T \triangle S \) must dominate the equation. Hence, for \(\triangle G\) to be negative at high \(T\), \( \triangle S \) should be positive because this makes \( -T \triangle S \) a large negative value.

Determine Sign of \(\triangle H \)

With \(\triangle S \) being positive, \(\triangle H \) needs to be positive to ensure that at low temperatures \(\triangle G\) can still be positive (reaction is non-spontaneous), but at high temperatures, \(-T \triangle S \) becomes more negative and \(\triangle G \) turns negative, making the reaction spontaneous.

Provide an Example Process

Consider the melting of ice to water. This process is endothermic (\(\triangle H > 0\)) and increases disorder (\(\triangle S > 0\)). Ice melts spontaneously at temperatures above 0°C because of the high positive entropy change.

Key concepts

Gibbs free energy

Gibbs free energy, often denoted as \(\triangle G\), is a thermodynamic quantity that can predict whether a reaction will be spontaneous. The formula to calculate Gibbs free energy is: \[ \triangle G = \triangle H - T \triangle S \] Here, \(\triangle H\) represents the change in enthalpy (heat content) of a system, \(\triangle S\) is the change in entropy (disorder), and \(\text{T}\) is the absolute temperature in Kelvin.

When \(\triangle G\) is negative, the reaction is spontaneous, meaning it can proceed without any input of energy. If \(\triangle G\) is positive, the reaction is non-spontaneous and requires external energy. Finally, when \(\triangle G\) is zero, the system is at equilibrium, and no net change occurs.

For a reaction that is only spontaneous at high temperatures, the term \(- T \triangle S\) must dominate. This happens when entropy change \(\triangle S\) is positive, making \(- T \triangle S\) a large negative value at high \(\text{T}\).

spontaneity of reactions

Spontaneity of reactions refers to whether a reaction occurs naturally without external intervention. It depends on Gibbs free energy, \(\triangle G\). For a reaction to be spontaneous, \(\triangle G\) must be negative.

At high temperatures, the spontaneity is influenced by the entropy change • \(\triangle S\), and enthalpy change • \(\triangle H \). For instance, if a reaction has a positive \(\triangle S\) (increase in disorder) and positive \(\triangle H\) (endothermic process), it would only be spontaneous at high temperatures.

In such cases, the term \(- T \triangle S\) becomes more negative as temperature • \(\text{T}\) increases, eventually making \(\triangle G\) negative. An example is the melting of ice. At low temperatures, the reaction is non-spontaneous, but as temperature rises, it becomes spontaneous due to the high positive entropy change.

entropy and enthalpy

Entropy \(\triangle S\) is a measure of the disorder or randomness in a system. A positive entropy change \(\triangle S > 0 \) indicates an increase in disorder. Enthalpy \(\triangle H\) represents the heat content of a system. If \(\triangle H\) is positive, the reaction is endothermic (absorbs heat); if \(\triangle H\) is negative, the reaction is exothermic (releases heat).

A reaction that is spontaneous only at high temperatures often has both \(\triangle S > 0 \) and \(\triangle H > 0\). The high positive entropy change (e.g., melting of ice) requires more heat to overcome the endothermic nature of the reaction.

This means at high temperatures, the large negative value of \(- T \triangle S\) can offset the positive \(\triangle H\), resulting in a negative \(\triangle G\), making the reaction spontaneous. At low temperatures, \(\triangle G\) can remain positive, showing the reaction is non-spontaneous under such conditions.