Question

For a chemical reaction with \(K<1\), is \(\Delta G \mathrm{~T}\) positive or negative?

Short answer

For a chemical reaction with \(K<1\), \(\Delta G \mathrm{~T}\) is positive.

Step-by-step solution

Understand the Equation for \(\Delta G \mathrm{~T}\)

Before we can solve, you need to understand the equation that links \(K\) and \(\Delta G \mathrm{~T}\). The equation is \(\Delta G \mathrm{~T} = -RT \ln K\), where \(R\) is the universal gas constant, \(T\) is the absolute temperature, and \(\ln\) denotes the natural logarithm.

Analyze the Sign of \(\Delta G \mathrm{~T}\)

Since \(R\) and \(T\) are always positive, the sign of \(\Delta G \mathrm{~T}\) is determined by the value of \(-\ln K\). Remember that the natural logarithm of a number less than 1 is negative, and when multiplied by \(-1\), the result is positive. Therefore, for a chemical reaction with \(K<1\), \(\Delta G \mathrm{~T}\) is positive.

Understand the Meaning of Positive \(\Delta G \mathrm{~T}\)

It's also important to understand what a positive \(\Delta G \mathrm{~T}\) means. A positive \(\Delta G \mathrm{~T}\) means the reaction is not spontaneous. It requires energy to move forward. In the context of the reaction quotient \(K\), \(K<1\) means the reaction has more reactants than products, hence it is not in equilibrium and will not proceed spontaneously in the forward direction.

Key concepts

Gibbs Free Energy

In the world of chemical thermodynamics, Gibbs Free Energy (\( \Delta G \)) helps us predict if a reaction can occur spontaneously. It's a handy tool that combines enthalpy (heat content) and entropy (disorder) into one value.

The equation is:\[\Delta G = \Delta H - T \Delta S\]

• \(\Delta H\) is the change in enthalpy.
• \(T\) is the temperature in Kelvin.
• \(\Delta S\) is the change in entropy.

If \( \Delta G \) is negative, the reaction releases energy and is spontaneous. A positive \( \Delta G\) means the reaction needs energy to proceed.

The relationship with the equilibrium constant (\( K \)) is captured by the equation:\[\Delta G = -RT \ln K\]In this equation, if \(K < 1\), then \(\ln K\) is negative, making \(-RT \ln K\) positive. This relates to the idea that the reaction is not spontaneous when there are more reactants than products.

Reaction Spontaneity

Understanding reaction spontaneity is crucial in predicting whether a chemical reaction will proceed on its own.

Spontaneous reactions tend to happen naturally without external input. These are marked by a negative Gibbs Free Energy (\( \Delta G \)).

Why does this matter?

• A spontaneous process often results in an increase in entropy or release of heat.
• Non-spontaneous reactions need energy, often as heat or work, to proceed.

For instance:

• Combustion of gasoline is spontaneous, releasing energy and increasing disorder.
• However, ice melting at cold temperatures is not spontaneous without adding energy.

If a reaction's \( \Delta G \) is positive, it tells us that the reaction needs an input of energy to occur. This echoes our earlier understanding: when \(K<1\), the concentration of reactants is higher than that of products, leading to non-spontaneity.

Equilibrium Constant

The equilibrium constant (\( K \)) offers insight into the concentration of reactants and products at equilibrium.

It’s a measure of the reaction's balance point where forward and reverse reactions occur at the same rate. A high \( K \) value means products are favored at equilibrium, whereas a low \( K \) value indicates a preference for reactants.

When \( K < 1\), it suggests that there are more reactants than products at equilibrium. This is a key insight for evaluating reaction direction and spontaneity.

• A small \(K\) means \( \ln K \) is negative.
• Consequently, \(-RT \ln K\) becomes positive, giving \(\Delta G\) a positive value.

This aligns with non-spontaneous reactions, as more reactants linger and less product forms naturally. Understanding this helps predict the necessary conditions to tip the balance towards product formation, often by adding energy.