Question

The standard free energy change of a reaction is \(\Delta G^{\circ}=-115 \mathrm{~kJ}\) at \(298 \mathrm{~K}\). Calculate the value of \(\log _{10} K_{p}\left(R=8.314 \mathrm{JK}^{-1} \mathrm{~mol}^{-1}\right)\) (a) \(20.16\) (b) \(2.303\) (c) \(2.016\) (d) \(13.83\)

Short answer

The value of \(\log_{10} K_{p}\) is approximately 19.14.

Step-by-step solution

Understand the relationship between free energy and equilibrium constant

The relationship between the standard free energy change \(\Delta G^\circ\) and the equilibrium constant \(K_p\) of a reaction at a temperature \(T\) is given by the equation \[\Delta G^\circ = -RT\ln(K_p)\], where R is the universal gas constant and T is the temperature in Kelvin.

Convert the natural logarithm to log base 10

The given equation can be expressed in terms of log base 10 using the relationship \[\ln(K_p) = \log_{10}(K_p)\ln(10)\]. Hence, the equation becomes \[\Delta G^\circ = -RT\log_{10}(K_p)\ln(10)\].

Isolate \(\log_{10}(K_p)\)

Rearrange the equation to solve for \(\log_{10}(K_p)\), \[\log_{10}(K_p) = -\frac{\Delta G^\circ}{RT\ln(10)}\].

Calculate \(\log_{10}(K_p)\) using the given values

Substitute the given values into the equation \(\Delta G^\circ = -115,000 \text{J}, R = 8.314 \text{JK}^{-1}\text{mol}^{-1}, T = 298 \text{K}\), and \(\ln(10)\approx 2.303\) to get \[\log_{10}(K_p) = -\frac{-115,000 \text{J}}{(8.314 \text{JK}^{-1}\text{mol}^{-1})(298 \text{K})(2.303)}\].

Perform the calculation

Carry out the calculation to find \(\log_{10}(K_p)\), \[\log_{10}(K_p) = -\frac{-115,000}{(8.314)(298)(2.303)}\]. \[\log_{10}(K_p) \approx 19.14\]. As the logarithm to base 10 of \(K_p\) should be a positive value, we correct the sign to be positive, concluding \[\log_{10}(K_p) = 19.14\].

Key concepts

Equilibrium Constant

The equilibrium constant, denoted as K or K_eq, is a profound concept in chemistry that quantifies the relative amounts of products and reactants at equilibrium in a chemical reaction. In a balanced chemical equation, the equilibrium constant is the ratio of the concentrations of products to reactants, each raised to the power of their stoichiometric coefficients. Understanding this ratio is crucial since it tells us whether a reaction favors the formation of products (when K > 1) or reactants (when K < 1).

The equilibrium constant is particularly valuable because it is related to the standard free energy change of the reaction. The standard free energy change (ΔG^°) is a measure of the driving force behind a chemical reaction. A negative ΔG^° indicates a spontaneous process, meaning the reaction will proceed in the forward direction under standard conditions. With the equation ewline ΔG^° = -RT&ln;(K_p),

we can calculate K_p, the equilibrium constant for gas-phase reactions based on partial pressures, using the free energy. By re-arranging this equation, as shown in the step-by-step solution, we are able to solve for the log₁₀(K_p) and thus understand the extent to which a reaction will proceed.

Reaction Spontaneity

The spontaneity of a chemical reaction is a key aspect of thermodynamics that tells us whether a reaction will occur without the need for additional energy input. A reaction is considered spontaneous if it could proceed on its own, once initiated, under a given set of conditions without any external intervention. In thermodynamics, we quantify spontaneity using the Gibbs free energy change (ΔG).

A negative value of ΔG indicates a spontaneous process, while a positive value suggests non-spontaneity. At equilibrium, ΔG is zero because there is no longer a net change occurring in the system. The relationship between ΔG and equilibrium constants, as demonstrated in the exercise, is powerful because it links the direction and extent of a reaction to a measurable quantity, ΔG^°. By calculating ΔG^°, one can predict whether a process is energetically favorable and use this to assess the potential of a reaction to do work or produce energy.

Thermodynamics in Chemistry

Thermodynamics is the branch of physical chemistry that deals with the relationships between heat, work, temperature, and energy. In the context of chemical reactions, thermodynamics provides a set of tools for predicting whether a reaction can occur and what changes will take place. The first and second laws of thermodynamics, in particular, play a significant role in understanding chemical processes.

The first law of thermodynamics, also known as the law of conservation of energy, states that energy cannot be created or destroyed in an isolated system. This means that the total energy of the universe remains constant, and during a chemical reaction, energy can only be transferred or transformed.

The second law of thermodynamics introduces the concept of entropy, a measure of the disorder or randomness in a system. It states that for any spontaneous process, the entropy of the universe will increase. This law helps chemists understand that not only must energy considerations (enthalpy) be favorable for a reaction, but entropy considerations often play a crucial part in determining reaction spontaneity.

The exercise illustrates the practical application of thermodynamic principles by using the Gibbs free energy equation to determine the reaction spontaneity and equilibrium constant. It's an example of how theoretical concepts are applied to calculate and predict the behavior of chemical systems.