Question

(a) If \(K<<1\) for a reaction, what do you know about the sign and magnitude of \(\Delta G^{\circ} ?\) (b) If \(\Delta G^{\circ}>>0\) for a reaction, what do you know about the magnitude of \(K ?\) Of \(Q ?\)

Short answer

If \ K <<1\), \(\backslash\backslashDelta G^{\circ}\) is large and positive. If \ \Delta G^{\circ} >> 0 \, K is very small, and \(\backslash\backslash Q\) is likely very small too.

Step-by-step solution

Understand the Relationship Between \(\backslash\backslashDelta G^{\circ}\) and K

Using the equation \( \Delta G^{\circ} = -RT \ln K \), we can see the relationship between \(\backslash\backslashDelta G^{\circ}\) and the equilibrium constant \(K\). Because \(R\) (the gas constant) and \(\backslash\backslashT\) (temperature in Kelvin) are both positive constants, the sign and magnitude of \(\backslash\backslashDelta G^{\circ}\) depend on \(\backslash\backslashln K\).

Analyze the Case When \( K

If \( K << 1 \), this implies that the reaction heavily favors the reactants and there are very few products at equilibrium. Since \(K\) is very small, \( \ln K \) will be a large negative number. Substituting this into the equation \( \Delta G^{\circ} = -RT \ln K \), a large negative \( \ln K \) will result in a large positive \( \Delta G^{\circ} \). Therefore, \(\backslash\backslashDelta G^{\circ}\) will be positive and have a large magnitude.

Evaluate \(\backslash\backslashDelta G^{\circ}

For a reaction with \( \Delta G^{\circ} >> 0 \), this implies a large positive standard Gibbs free energy. We already know from the previous step that for \( \Delta G^{\circ} \) to be large and positive, \(\backslash\backslash ln K\) must be large and negative, meaning \( K \) must be very small (\backslash\backslash K << 1\backslash).

Relate \( Q \) to \( \Delta G^{\circ} \)

The reaction quotient \(\backslash\backslash Q\) is similar to \( K \) but is calculated with initial concentrations instead of equilibrium concentrations. If \( \Delta G^{\circ} \) is significantly greater than zero, and since \( Q \) is often around the initial value comparable to \( K \) at reaction start, it follows that \(\backslash\backslash Q\) should be small, likely close to the small value of \( K \). Therefore, \(\backslash\backslash Q \) is also likely to be much smaller than 1.

Key concepts

standard Gibbs free energy (ΔG^{\textdegree})

To understand the standard Gibbs free energy (\textDelta G^{\textdegree}), it's crucial to know that it represents the maximum amount of work a system can perform at constant temperature and pressure. This is given by the equation:
\
\[ \textDelta G^{\textdegree} = -RT \textIndicator {ln} K \]

where \R\ (gas constant) and \T\ (temperature in Kelvin) are constants.

\textDelta G^{\textdegree} provides insights into the feasibility and spontaneity of a reaction:

• If \textDelta G^{\textdegree} < 0: The reaction is spontaneous.
• If \textDelta G^{\textdegree} > 0: The reaction is non-spontaneous.

In scenarios where
\text K << 1: There are far more reactants than products at equilibrium, resulting in \textIndicator {ln} K\ being a large negative value. Thus, \textΔ G^{\textdegree}\ is large and positive.

Conversely, if \textΔG^{\textdegree}\ is quite large and positive, \text K\ must be quite small, indicating very few products at equilibrium.

equilibrium constant (K)

The equilibrium constant (K) is pivotal in chemical thermodynamics, as it indicates the ratio of product concentrations to reactant concentrations at equilibrium. It is calculated at a constant temperature.

• When K > 1: Products are favored at equilibrium.
• When K < 1: Reactants are favored at equilibrium.

The relationship between K and \text∆G^{\textdegree}\ is defined by the formula:

\[ \textDelta G^{\textdegree} = -RT \textIndicator {ln} K \]

This equation elucidates the inverse relationship between \text∆G^{\textdegree}\ and K:

• If \text K << 1\, then \textIndicator {ln} K\ is large and negative, leading to a large and positive \text∆G^{\textdegree}\.
• If \text K >> 1\, then \textIndicator {ln} K\ is large and positive, making \text∆G^{\textdegree}\ large and negative.

reaction quotient (Q)

The reaction quotient (\text Q) is similar to the equilibrium constant (K) but reflects the ratio of product concentrations to reactant concentrations at any point in time, not necessarily at equilibrium.

Just like K, Q is calculated using initial or current concentrations:

\[ Q = \frac { [Products]_{current} } { [Reactants]_{current} } \]

By comparing Q with K, we can predict the direction of the reaction to reach equilibrium:

• If Q < K: The reaction proceeds forward (toward products).
• If Q > K: The reaction proceeds backward (toward reactants).
• If Q = K: The system is already at equilibrium.

For a reaction with significantly positive \textΔG^{\textdegree}\, both K and Q are typically small, i.e., << 1. This condition implies that the reaction, even initially or during progression, heavily favors the reactants over time.