Chapter 6: Problem 98
The equilibrium constant for a reaction is \(10 .\) What will be the value of \(\Delta G^{\circ}\) at \(300 \mathrm{~K} ?\) (a) \(-5.74 \mathrm{~kJ}\) (b) \(-574 \mathrm{~kJ}\) (c) \(+11.48 \mathrm{~kJ}\) (d) \(+5.74 \mathrm{~kJ}\)
Short Answer
Expert verified
-5.74 kJ/mol
Step by step solution
01
Writing the formula to calculate \(\Delta G^{\circ}\)
Determine the Gibbs free energy change \(\Delta G^{\circ}\) for the reaction at equilibrium by using the formula: \[\Delta G^{\circ} = -RT \ln K\] where \(R\) is the universal gas constant, \(T\) is the temperature in Kelvin, and \(K\) is the equilibrium constant.
02
Plug in the values
Use the given values to calculate \(\Delta G^{\circ}\). Here, \(K = 10\), \(T = 300\ K\), and the universal gas constant \(R = 8.314\ J/(mol\cdot K)\). Convert \(R\) to kJ by dividing by 1000 if you wish to get your answer in kJ. The equation becomes: \[\Delta G^{\circ} = -\left( \frac{8.314\ J}{mol\cdot K} \right)\left(300\ K\right)\ln(10)\]
03
Calculate \(\Delta G^{\circ}\)
Perform the calculation. Keeping in mind \(\ln(10) \approx 2.303\), \[\Delta G^{\circ} = -\left(\frac{8.314}{1000}\frac{kJ}{mol\cdot K}\right)\left(300\ K\right)(2.303) = -\left(\frac{8.314}{1000}\right)(300)(2.303)\approx -5.74\ kJ/mol\]
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Understanding Equilibrium Constants
When studying chemical reactions, the notion of equilibrium is central. At equilibrium, the rate of the forward reaction equals the rate of the reverse reaction, resulting in no net change in the concentrations of products and reactants over time. The equilibrium constant (K) represents the ratio of product concentrations to reactant concentrations at this state, each raised to the power of their respective coefficients in the balanced equation.
The magnitude of K provides insight into the position of equilibrium. A large K (much greater than 1) implies a reaction favoring products; conversely, a small K (much less than 1) suggests that reactants are favored. Understanding this concept is crucial for predicting the extent to which a reaction will proceed under specific conditions.
The magnitude of K provides insight into the position of equilibrium. A large K (much greater than 1) implies a reaction favoring products; conversely, a small K (much less than 1) suggests that reactants are favored. Understanding this concept is crucial for predicting the extent to which a reaction will proceed under specific conditions.
Calculating K from Gibbs Free Energy
In the context of the exercise, knowing K allows for the determination of the Gibbs free energy change, Delta G°. The relationship between these values is governed by the equation Delta G° = -RT ln(K) where R is the universal gas constant, T is the temperature in Kelvin, and ln(K) is the natural logarithm of the equilibrium constant. Grasping the correlation between Delta G° and K is imperative to predict the spontaneity and directionality of a reaction.Exploring Chemical Thermodynamics
Chemical thermodynamics involves the study of energy changes accompanying chemical reactions and physical processes. It paves the way for predicting reactant and product behavior under varying conditions of temperature and pressure. One central concept is Gibbs free energy (Delta G°), which represents the maximum amount of work that can be performed by a system at constant temperature and pressure.
The provided exercise exemplifies a common thermodynamics problem—calculating Delta G° from a known equilibrium constant to ascertain reaction spontaneity. Comprehending these thermodynamic principles is instrumental in various fields, from developing new materials to understanding biological systems.
Significance of Gibbs Free Energy
Delta G° is a critical indicator of reaction spontaneity; negative values suggest a reaction is spontaneous, while positive values indicate non-spontaneity. When Delta G° equals zero, the system is at equilibrium, and no net work can be done. By calculating Delta G°, you unearth whether a reaction can occur without external energy input.The provided exercise exemplifies a common thermodynamics problem—calculating Delta G° from a known equilibrium constant to ascertain reaction spontaneity. Comprehending these thermodynamic principles is instrumental in various fields, from developing new materials to understanding biological systems.
Determining Reaction Spontaneity
A spontaneous reaction is one that occurs on its own, without external intervention. However, spontaneity does not equate to speed; rather, it refers to the inherent tendency of a process to occur under a given set of conditions. To determine this tendency, chemists assess the Gibbs free energy change (Delta G°).
A key takeaway is that spontaneity is direction-dependent. For a given reaction occurring at a particular temperature and pressure, if Delta G° is negative, the forward reaction is spontaneous. If Delta G° is positive, the reverse reaction is spontaneous.
When approaching problems such as the one in our exercise, a solid understanding of these energy concepts is invaluable. It allows you to predict if a reaction will occur as written (forward direction) or if the system will favor the reactants (reverse direction), straight from the numeric value of Delta G°. The clear interpretation of Delta G° in conjunction with the equilibrium constant provides a full picture of the reaction's characteristics and potential to proceed.
A key takeaway is that spontaneity is direction-dependent. For a given reaction occurring at a particular temperature and pressure, if Delta G° is negative, the forward reaction is spontaneous. If Delta G° is positive, the reverse reaction is spontaneous.
Understanding 'Spontaneity' in Thermodynamic Terms
What does this mean in practical terms? A negative Delta G° suggests that a reaction could theoretically initiate and proceed without any external energy—this is often associated with liberating energy (exergonic reaction). On the other hand, a positive Delta G° would require an input of energy to drive the reaction (endergonic reaction).When approaching problems such as the one in our exercise, a solid understanding of these energy concepts is invaluable. It allows you to predict if a reaction will occur as written (forward direction) or if the system will favor the reactants (reverse direction), straight from the numeric value of Delta G°. The clear interpretation of Delta G° in conjunction with the equilibrium constant provides a full picture of the reaction's characteristics and potential to proceed.