Graphing Calculator Calculating the Gibbs-Energy Change The graphing calculator can run a program that calculates the Gibbs-energy change, given the temperature, \(T\) , change in enthalpy \(\Delta H,\) and change in entropy, \(\Delta S .\) Given that the temperature is 298 \(\mathrm{K}\) , the change in enthalpy is 131.3 \(\mathrm{kJ} / \mathrm{mol}\) , and the change in entropy is \(0.134 \mathrm{kJ} /(\mathrm{mol} \cdot \mathrm{K}),\) you can calculate Gibbs-energy change in kilojoules per mole. Then use the program to make calculations. Go to Appendix C. If you are using a TI-83 Plus, you can download the program ENERGY data and run the application as directed. If you are using another calculator, your teacher will provide you with keystrokes and data sets to use. After you have run the program, answer the following questions. \begin{equation} \begin{array}{l}{\text { a. What is the Gibbs-energy change given a }} \\\ {\text { temperature of } 300 \mathrm{K}, \text { a change in enthalpy }} \\\ {\text { of } 132 \mathrm{kJ} / \mathrm{mol} \text { and a change in entropy of }} \\ {0.086 \mathrm{kJ} / /(\mathrm{mol} \cdot \mathrm{K}) ?}\\\\{\text { b. What is the Gibbs-energy change given a }} \\ {\text { temperature of } 288 \mathrm{K}, \text { a change in enthalpy }} \\ {\text { of } 115 \mathrm{kJ} / \mathrm{mol} \text { and a change in entropy of }} \\ {0.113 \mathrm{kJ} /(\mathrm{mol} \cdot \mathrm{K}) ?}\\\\{\text { c. What is the Gibbs-energy change given a }} \\ {\text { temperature of } 298 \mathrm{K} \text { , a change in enthalpy }} \\ {\text { of } 181 \mathrm{kJ} / \mathrm{mol} \text { and a change in entropy of }} \\ {0.135 \mathrm{kJ} /(\mathrm{mol} \cdot \mathrm{K}) ?}\end{array} \end{equation}

Step by step solution

01

Understand the Equation

The Gibbs-Helmholtz equation for calculating the change in Gibbs free energy is as follows: \[\Delta G= \Delta H- T\Delta S\] Where \(\Delta G\) is the change in Gibbs free energy, \(\Delta H\) is the change in enthalpy, \(T\) is the temperature and \(\Delta S\) is the change in entropy.

02

Apply the Equation for Each Scenario

a. Use the provided values in the equation given that \(T=300\) Kelvin, \(\Delta H=132\)kJ/mol and \(\Delta S=0.086\)kJ/(mol·K) to get: \[\Delta G = 132 \ kJ/mol - 300K * 0.086 \ kJ/(mol * K)\] b. For the second scenario with given \(T=288\) Kelvin, \(\Delta H=115 \)kJ/mol and \(\Delta S=0.113 \)kJ/(mol·K), plug these into the equation for \(\Delta G\): \[\Delta G = 115 \ kJ/mol - 288K * 0.113 \ kJ/(mol * K)\]c. In the last scenario, use values \(T=298 \)Kelvin, \(\Delta H=181 \)kJ/mol and \(\Delta S=0.135 \)kJ/(mol·K) to compute: \[\Delta G = 181 \ kJ/mol -298K * 0.135 \ kJ/(mol * K) \]

03

Solve for the Gibbs Energy Change

Solve each equation for \(\Delta G\) to find the Gibbs free energy change for the system in each case. Remember converting \(kJ\) into \(J\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Gibbs-Helmholtz Equation

The Gibbs-Helmholtz Equation is a powerful formula used in chemistry and physics to predict the spontaneity of a process at constant temperature and pressure. The equation is given by:
\[\Delta G = \Delta H - T\Delta S\] where \(\Delta G\) represents the change in Gibbs free energy, \(\Delta H\) is the change in enthalpy, \(T\) is the absolute temperature in Kelvin (K), and \(\Delta S\) is the change in entropy.

Essentially, the equation expresses that the change in Gibbs free energy is the total heat content or enthalpy of a system minus the product of temperature and the change in entropy. When \(\Delta G\) is negative, the process is spontaneous, leading towards equilibrium. For a positive \(\Delta G\), the process requires energy input to occur, hence non-spontaneous. When \(\Delta G\) is zero, the system is in a state of equilibrium. Understanding how to use this formula is critical for predicting the direction and extent of chemical reactions.

Enthalpy Change

Enthalpy change, denoted as \(\Delta H\), is a key concept in thermodynamics related to the heat content within a chemical system under constant pressure. It can be thought of as the total energy within a system, which includes internal energy and the energy required to displace the system's surroundings.

A negative \(\Delta H\) indicates that a reaction is exothermic, meaning it releases heat to the surroundings. Conversely, a positive \(\Delta H\) means the reaction is endothermic and absorbs heat. Calculating the enthalpy change for a process relies on understanding the substances involved, their states, and how they interact under specific conditions. In the context of Gibbs-Helmholtz equation, the enthalpy change is a critical component to determine the spontaneity of a reaction.

Entropy Change

Entropy change, symbolized as \(\Delta S\), reflects the degree of randomness or disorder within a system. It's a fundamental thermodynamic quantity that is often discussed alongside energy and enthalpy changes. When a system undergoes a change, such as a chemical reaction, its entropy can increase or decrease.

An increase in entropy (\(\Delta S > 0\)) means the system's disorder has increased. This is common in reactions where the number of particles increases, such as when a solid melts or a liquid vaporizes. On the other hand, a decrease in entropy (\(\Delta S < 0\)) indicates a transition towards order, like when a gas condenses to a liquid.

In the Gibbs-Helmholtz equation, \(T\Delta S\) represents the energy 'spread' through the increase or decrease in entropy across a temperature, directly affecting the Gibbs free energy change and thus the feasibility of a process.

Graphing Calculator Applications

Graphing calculators are versatile tools that can be applied to various scientific calculations, including Gibbs-energy change determination. Known for their computational power, these devices can execute programs or applications that streamline complex processes. For instance, students can use a graphing calculator to input the temperature (T), change in enthalpy (\(\Delta H\)), and change in entropy (\(\Delta S\)) directly to quickly calculate the Gibbs-energy change (\(\Delta G\)).

This capability is incredibly useful in educational settings, where students are exploring thermodynamic concepts and need to evaluate the spontaneity and feasibility of reactions under various conditions. With appropriate programs installed, graphing calculators can save time, reduce the potential for manual error, and help solidify understanding of the underlying principles of thermodynamics.