Problem 56 - Chemistry: Principles and Reactions

Question

Show by calculation whether the reaction $$ \mathrm{HC}_{2} \mathrm{H}_{3} \mathrm{O}_{2}(a q) \rightleftharpoons \mathrm{H}^{+}(a q)+\mathrm{C}_{2} \mathrm{H}_{3} \mathrm{O}_{2}^{-}(a q) \quad \Delta G^{\circ}=+27.2 \mathrm{~kJ} $$ is spontaneous at \(25^{\circ} \mathrm{C}\) (a) when \(\left[\mathrm{H}^{+}\right]=\left[\mathrm{C}_{2} \mathrm{H}_{3} \mathrm{O}_{2}^{-}\right]=0.85 M_{;}\left[\mathrm{HC}_{2} \mathrm{H}_{3} \mathrm{O}_{2}\right]=0.15 \mathrm{M}\). (b) when \(\left[\mathrm{H}^{+}\right]=\left[\mathrm{C}_{2} \mathrm{H}_{3} \mathrm{O}_{2}^{-}\right]=2.0 \times 10^{-3} \mathrm{M} ;\left[\mathrm{HC}_{2} \mathrm{H}_{3} \mathrm{O}_{2}\right]=1.0 \mathrm{M}\).

Step-by-Step Solution

Answer

Answer: The reaction is not spontaneous under the conditions in case (a) and is spontaneous under the conditions in case (b).

Step 1: Calculate Q

(Write the content here) We calculate the reaction quotient Q using the given concentrations: $$ Q =\frac{[\mathrm{H}^{+}][\mathrm{C}_{2} \mathrm{H}_{3} \mathrm{O}_{2}^{-}]}{[\mathrm{HC}_{2} \mathrm{H}_{3} \mathrm{O}_{2}]} $$ Plugging in the concentrations: $$ Q =\frac{(0.85)(0.85)}{(0.15)}=4.816 $$

Step 2: Calculate ΔG

Now, we calculate the Gibbs free energy change under non-standard conditions using the formula: $$ ΔG = ΔG° + RT \ln(Q) $$ Plugging in the values: $$ ΔG = 27,200 + (8.314)(298.15)\ln(4.816)=23,980.6 \mathrm{~ J} $$

Step 3: Determine if the reaction is spontaneous

Since ΔG > 0 (23,980.6 J), the reaction is not spontaneous under these conditions. ##Case (b)##

Step 1: Calculate Q

We calculate the reaction quotient Q using the given concentrations: $$ Q =\frac{[\mathrm{H}^{+}][\mathrm{C}_{2} \mathrm{H}_{3} \mathrm{O}_{2}^{-}]}{[\mathrm{HC}_{2} \mathrm{H}_{3} \mathrm{O}_{2}]} $$ Plugging in the concentrations: $$ Q =\frac{(2.0 \times 10^{-3})(2.0 \times 10^{-3})}{(1.0)}=4.0 \times 10^{-6} $$

Step 2: Calculate ΔG

Now, we calculate the Gibbs free energy change under non-standard conditions using the formula: $$ ΔG = ΔG° + RT \ln(Q) $$ Plugging in the values: $$ ΔG = 27,200 + (8.314)(298.15)\ln(4.0 \times 10^{-6})=-6743.1 \mathrm{~ J} $$

Step 3: Determine if the reaction is spontaneous

Since ΔG < 0 (-6743.1 J), the reaction is spontaneous under these conditions. In conclusion, the given reaction is not spontaneous for case (a) but is spontaneous for case (b).

Key Concepts

Chemical Spontaneity

Chemical spontaneity is a term in chemistry that helps us understand whether a reaction can occur without external assistance. It does not necessarily mean 'instantly'; Rather, it refers to whether a reaction will proceed under a given set of conditions. The energetics of chemical reactions are governed by Gibbs free energy, represented by the symbol \( \Delta G \).

Gibbs free energy is a measure of the total energy available to do work during a chemical reaction. If \( \Delta G \), for a process is negative, the reaction is said to be spontaneous; it can proceed forward and release energy. Conversely, a positive \( \Delta G \) indicates that the reaction is non-spontaneous; it requires energy to proceed. It's important to note that spontaneous reactions may still be slow; spontaneity doesn't imply speed, just the thermodynamic favorability of the reaction.

Reaction Quotient (Q)

The reaction quotient (\( Q \)) is a key concept in understanding reactions that are not at equilibrium. It is calculated by taking the ratio of the concentrations of the products raised to their stoichiometric coefficients, to the concentrations of the reactants raised to their stoichiometric coefficients, at a given moment in time.

\( Q \) is compared with the equilibrium constant (\( K \)) to predict which direction a reaction will proceed to reach equilibrium.

• If \( Q=K \) the reaction is at equilibrium.
• If \( Q
• If \( Q>K \) the reaction will proceed in the reverse direction, converting products back into reactants.

The concept of \( Q \) helps in calculating the Gibbs free energy change (\( \Delta G \) ) in reactions that are not at equilibrium by using the formula \( \Delta G = \Delta G^\circ + RT \ln(Q) \).

Equilibrium Constant

The equilibrium constant, represented by \( K \) for a chemical reaction, is a dimensionless number that provides a measure of the position of the reaction at equilibrium. It is calculated similarly to the reaction quotient (\( Q \)), but at the conditions when the reaction has reached equilibrium — that is, when the rates of the forward and reverse reactions are equal.

The magnitude of \( K \) can indicate the extent to which a reaction will proceed:

• A large \( K \) (>1) means that at equilibrium, the products are favored over the reactants.
• A small \( K \) (<1) indicates that reactants are favored.

This information is essential in predicting how the reaction mixture will respond to changes in conditions, according to Le Chatelier's principle, and in calculating standard Gibbs free energy change using the equation \( \Delta G^\circ = -RT \ln(K) \).

Thermodynamics in Chemistry

Thermodynamics is the branch of physical chemistry that deals with the energy changes associated with chemical reactions and the direction in which processes occur. It rests upon three fundamental laws, plus the concept of entropy. In the context of chemical reactions, the second law of thermodynamics states that the total entropy of a closed system can never decrease over time. This means that processes naturally tend to move towards a state of disorder or randomness, which relates to the concept of spontaneous reactions.

The major player in thermodynamics within chemistry is the Gibbs free energy, which combines enthalpy (the heat content of a system), entropy (the degree of disorder), and temperature to predict the spontaneity of reactions. Understanding thermodynamics allows chemists to control conditions to drive non-spontaneous reactions, understand reaction energetics, and manipulate equilibrium positions to favor the formation of desired products.