Problem 74 - Advanced Problems in Physical Chemistry for Competitive Examinations

Question

An amount of 5 mole \(\mathrm{H}_{2} \mathrm{O}(1)\) at \(100^{\circ} \mathrm{C}\) and \(1 \mathrm{~atm}\) is converted into \(\mathrm{H}_{2} \mathrm{O}(\mathrm{g})\) at \(100^{\circ} \mathrm{C}\) and 5 atm. \(\Delta G\) for the process is (a) zero (b) \(1865 \ln 5 \mathrm{cal}\) (c) \(3730 \ln 5 \mathrm{cal}\) (d) \(-3730 \ln 5 \mathrm{cal}\)

Step-by-Step Solution

Answer

The Gibbs free energy change (\(\Delta G\)) for the process is \(1865 \ln 5 \text{ cal}\).

Step 1 - Understand the Process

The problem describes the conversion of water from liquid to gas phase at constant temperature (100°C), but under a change in pressure from 1 atm to 5 atm. According to the Gibbs phase rule, when a pure substance is at its boiling point, the process of phase change is reversible and isothermal, meaning \( \Delta G \) for the phase change is zero at equilibrium.

Step 2 - Apply the Gibbs Free Energy Change Equation

The Gibbs free energy change \( \Delta G \) for a process when pressure changes at constant temperature can be given by the equation \( \Delta G = nRT \ln\left(\frac{P_2}{P_1}\right) \) where \( n \) is the number of moles, \( R \) is the gas constant, \( T \) is the temperature in Kelvin, \( P_1 \) and \( P_2 \) are the initial and final pressures respectively. Here, \( T = 273.15 + 100 \) K, \( R = 1.987 \text{ cal/mol}\cdot\text{K} \) (since the answer choices are in calories), \( n = 5 \) moles, \( P_1 = 1 \) atm and \( P_2 = 5 \) atm.

Step 3 - Calculate the Temperature in Kelvin

The temperature in degrees Celsius must be converted to Kelvin. \( T_K = T_{\text{°C}} + 273.15 \). For a temperature of \( 100^\circ\text{C} \) the Kelvin temperature is \( T_K = 100 + 273.15 = 373.15 \text{ K} \) .

Step 4 - Plug Values into the Equation

Substitute the known values into the Gibbs free energy change equation to calculate \( \Delta G \) : \(\Delta G = (5 \text{ mol}) (1.987 \text{ cal/mol}\cdot\text{K}) (373.15 \text{ K}) \ln\left(\frac{5 \text{ atm}}{1 \text{ atm}}\right) \) .

Step 5 - Calculate the Gibbs Free Energy Change

After plugging in the values, the calculation of \( \Delta G \) is straightforward. Use the natural logarithm of 5, which is approximately 1.609, and complete the multiplication: \( \Delta G = 5 \times 1.987 \times 373.15 \times 1.609 \approx 1865 \text{ cal} \times \ln(5) \).

Key Concepts

Phase Change

A phase change is a physical process in which a substance changes from one state of matter to another, such as from a liquid to a gas. These changes usually occur at specific temperatures and pressures, often referred to as the substance's boiling or melting points. During such a transition, the substance absorbs or releases energy, but its temperature remains constant.

For example, when water boils at 100°C under standard atmospheric pressure (1 atm), it remains at 100°C throughout the conversion from liquid to vapor phase. This process is known as a 'phase transition', and multiple phases can coexist at equilibrium at this point. Understanding the specific conditions under which these changes occur is crucial for grasping the basics of physical chemistry and thermodynamics.

Thermodynamics

Thermodynamics is a branch of physics that deals with the energy and work of a system. It's fundamentally important in chemistry, particularly when studying phase changes and chemical reactions. This field defines several fundamental quantities, such as energy, temperature, and entropy, in order to describe thermodynamic systems.

The laws of thermodynamics dictate how and when energy conversions can occur, and they explain why certain processes are spontaneous while others are not. Gibbs Free Energy, abbreviated as \( G \), is a thermodynamic potential that can be used to calculate the maximum amount of work obtainable from a closed thermodynamic system at constant pressure and temperature. It is particularly helpful for predicting the direction of chemical reactions and phase changes.

Chemical Equilibrium

Chemical equilibrium is a state in which the forward and reverse reactions occur at the same rate, resulting in no overall change in the amounts of products and reactants. This concept is pivotal in the area of physical chemistry, as it defines the point at which a chemical reaction is balanced.

In relation to Gibbs Free Energy, a state of equilibrium corresponds to the condition where \( \Delta G = 0 \). At this state, the system's free energy is at its minimum under the given conditions. For a process such as phase change, reaching equilibrium means that the rates of evaporation and condensation (in the case of liquid to vapor) are equal, maintaining an unchanging distribution of phases over time. Understanding how to calculate changes in Gibbs Free Energy helps chemists to predict not only the direction in which a reaction will proceed but also how external conditions such as pressure and temperature will affect equilibrium.

Physical Chemistry

Physical chemistry is the study of how matter behaves on a molecular and atomic level and how chemical reactions occur. From our studies of phase changes to the calculations of Gibbs Free Energy, physical chemistry involves understanding the physical properties of molecules, how they combine, and how they interact with energy.

In our textbook example, we applied the principles of physical chemistry to calculate Gibbs Free Energy during a phase transition under non-standard conditions. The concepts of equilibrium, thermodynamics, and phase changes intertwine within this field to explain and predict the energetic and physical outcomes of chemical processes. Simplifying complex theories into understandable concepts is an integral part of physical chemistry and education in this field should empower students to feel confident in applying these principles in practical scenarios.