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Chapter 4: Problem 87

What is the free energy change, \(\Delta G\), when \(1.0\) mole of water at \(100^{\circ} \mathrm{C}\) and 1 atm pressure is converted in to steam at \(100^{\circ} \mathrm{C}\) and 1 atm pressure? (a) \(540 \mathrm{cal}\) (b) \(-9800 \mathrm{cal}\) (c) \(9800 \mathrm{cal}\) (d) \(0 \mathrm{cal}\)

Short Answer

Expert verified
The free energy change, \(\Delta G\), for the conversion of water to steam at \(100^\circ C\) and 1 atm is \(0 \mathrm{cal}\).

Step by step solution

01

Understanding the concept of Gibbs Free Energy

The Gibbs Free Energy change, \(\Delta G\), for a process is given by the equation \(\Delta G = \Delta H - T\Delta S\), where \(\Delta H\) is the enthalpy change, \(T\) is the temperature in Kelvin, and \(\Delta S\) is the entropy change. For a phase change such as water converting to steam at constant temperature and pressure, \(\Delta G\) is equal to zero.
02

Analyzing the conditions

The conversion of water to steam at \(100^\circ C\) and 1 atm pressure is an equilibrium process at the boiling point. At equilibrium, the Gibbs Free Energy change, \(\Delta G\), is zero because the process is reversible, and there is no net change in the free energy of the system.
03

Choose the correct answer

Given that \(\Delta G = 0\) under these conditions, the correct answer is \(0 \mathrm{cal}\).

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Key Concepts

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

Enthalpy Change

Enthalpy change, often symbolized as \( \Delta H \), is a measure of the total heat content of a system. It represents the heat absorbed or released during a chemical reaction or phase transition under constant pressure. A positive \( \Delta H \) signifies that heat is absorbed by the system, which is termed as an endothermic process, while a negative \( \Delta H \) indicates heat is released, referred to as an exothermic process. In the context of our exercise example, the phase transition of water to steam involves an increase in enthalpy, because energy is required to convert liquid water into gaseous steam at its boiling point.

Understanding \( \Delta H \) in Phase Transitions

When water boils at \(100^\circ \mathrm{C}\) and 1 atm, it absorbs a specific amount of heat known as the heat of vaporization. This heat is essential for breaking the intermolecular forces within the liquid, allowing the molecules to spread out and form steam. Yet, this enthalpy change does not affect the free energy change \( \Delta G \) at the equilibrium boiling point, since \( \Delta G = 0 \) for this process.

Entropy Change

Entropy change, represented as \( \Delta S \), is a crucial concept in thermodynamics that indicates the disorder or randomness of the particles within a system. A positive entropy change means that the randomness of the particles is increasing, typically seen when a substance transitions from a solid or liquid to a gas. Conversely, a negative entropy change suggests a decrease in disorder, such as when a gas condenses into a liquid.

Entropy and Phase Transitions

The transition from water to steam involves an increase in entropy because the gas phase has much more randomness and freedom of particle movement compared to the liquid phase. During the boiling process, the organized structure of liquid water breaks down, leading to increased molecular disorder—which corresponds to a positive \( \Delta S \).

Chemical Equilibrium

Chemical equilibrium occurs when a reversible chemical reaction is proceeding at the same rate in both the forward and reverse directions, resulting in no net change of the reactants and products over time. At equilibrium, the concentrations of all reactants and products remain constant, though not necessarily equal. The point at which this balance is achieved is heavily influenced by factors such as temperature, pressure, and concentration.

Equilibrium and Gibbs Free Energy

At chemical equilibrium, the Gibbs Free Energy change \( \Delta G \) is zero. This signifies that the system is at its lowest energy state and no further work can be done. In our textbook exercise, the conversion of water to steam at \(100^\circ \mathrm{C}\) and 1 atm is an equilibrium process, thereby making \( \Delta G \) equal to zero. This is an essential concept when analyzing boiling and other phase transitions at constant temperature and pressure.

Phase Transition

A phase transition is a transformation of a substance from one state of matter to another, such as from a solid to a liquid, liquid to gas, or vice versa. These transitions involve a change in the energy and organization of particles within a substance and are typically associated with energy transfer in the form of heat. Common types of phase transitions include melting, freezing, vaporization, condensation, sublimation, and deposition.

Critical Conditions for Phase Transitions

Specific conditions must be met for a phase transition to occur. For example, water will only boil and transition to steam at its boiling point, which is \(100^\circ \mathrm{C}\) at 1 atm pressure. At this point, additional heat increases the kinetic energy of water molecules to overcome intermolecular forces, leading to a phase change. In our exercise scenario, since the phase transition occurs at equilibrium, no free energy change takes place, and \( \Delta G = 0 \).

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Most popular questions from this chapter

One mole of a real gas is subjected to heating at constant volume from \(\left(P_{1}\right.\), \(V_{1}, T_{1}\) ) state to \(\left(P_{2}, V_{1}, T_{2}\right)\) state. Then it is subjected to irreversible adiabatic compression against constant external pressure of \(P_{3}\) atm, till the system reaches final state \(\left(P_{3}, V_{2}, T_{3}\right) .\) If the constant volume molar heat capacity of real gas is \(C_{\mathrm{V}}\), then the correct expression for \(\Delta H\) from State 1 to State 3 is (a) \(C_{\mathrm{V}}\left(T_{3}-T_{1}\right)+\left(P_{3} V_{1}-P_{1} V_{1}\right)\) (b) \(C_{\mathrm{V}}\left(T_{2}-T_{1}\right)+\left(P_{3} V_{2}-P_{1} V_{1}\right)\) (c) \(C_{\mathrm{y}}\left(T_{2}-T_{1}\right)+\left(P_{3} V_{1}-P_{1} V_{1}\right)\) (d) \(C_{\mathrm{P}}\left(T_{2}-T_{1}\right)+\left(P_{3} V_{1}-P_{1} V_{1}\right)\)

In the reversible adiabatic expansion of an ideal monoatomic gas, the final volume is 8 times the initial volume. The ratio of final temperature to initial temperature is (a) \(8: 1\) (b) \(1: 4\) (c) \(1: 2\) (d) \(4: 1\)

With what minimum pressure must a given volume of an ideal gas \((\gamma=1.4)\), originally at \(400 \mathrm{~K}\) and \(100 \mathrm{kPa}\), be adiabatically compressed in order to raise its temperature up to \(700 \mathrm{~K}\) ? (a) \(708.9 \mathrm{kPa}\) (b) \(362.5 \mathrm{kPa}\) (c) \(1450 \mathrm{kPa}\) (d) \(437.4 \mathrm{kPa}\)

The work done in the isothermal reversible expansion of argon gas at \(27^{\circ} \mathrm{C}\) from 41 to 161 was equal to 4200 cal. What is the amount of argon subjected to such an expansion? (Ar \(=40, \ln 4=1.4\) ) (a) \(5.0 \mathrm{~g}\) (b) \(20.0 \mathrm{~g}\) (c) \(200.0 \mathrm{~g}\) (d) \(48.1 \mathrm{~g}\)

What is the entropy change when \(3.6 \mathrm{~g}\) of liquid water is completely converted into vapours at \(373 \mathrm{~K} ?\) The molar heat of vaporization is \(40.85 \mathrm{~kJ} / \mathrm{mol}\). (a) \(218.9 \mathrm{~J} / \mathrm{K}\) (b) \(2.189 \mathrm{~J} / \mathrm{K}\) (c) \(21.89 \mathrm{~J} / \mathrm{K}\) (d) \(0.2189 \mathrm{~J} / \mathrm{K}\)
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