Problem 78 - Advanced Problems in Physical Chemistry for Competitive Examinations

Question

A definite mass of a monoatomic ideal gas at 1 bar and \(27^{\circ} \mathrm{C}\) expands against \(\begin{array}{llll}\text { vacuum } & \text { from } & 1.2 \mathrm{dm}^{3} & \text { to } & 2.4 \mathrm{dm}^{3} \text { . }\end{array}\) The change in free energy of the gas, \(\Delta G\), is \((R=0.08\) bar- \(\mathrm{L} / \mathrm{K}-\mathrm{mol}, \ln 2=0.7)\) (a) 0 (b) \(-64\) bar- 1 (c) \(+84 \mathrm{~J}\) (d) \(-84 \mathrm{~J}\)

Step-by-Step Solution

Answer

The change in free energy of the gas, \(\Delta G\), is 0.

Step 1: Understanding the Expansion Process

The gas expands against a vacuum, which means that it does so without any external pressure (free expansion). Since external pressure is zero, work done by the gas is also zero during the expansion.

Step 2: Using the First Law of Thermodynamics

According to the first law of thermodynamics, which states that \(\Delta U = Q - W\), where \(\Delta U\) is the change in internal energy, \(Q\) is the heat added to the system, and \(W\) is the work done by the system. In the case of free expansion, \(W = 0\). For an ideal gas, the internal energy is a function of temperature only, and since the temperature is constant, \(\Delta U = 0\). Thus \(Q = 0\) as well.

Step 3: Calculating the Change in Free Energy

The change in free energy \(\Delta G\) for an ideal gas at constant temperature and pressure can be determined by the equation \(\Delta G = \Delta H - T\Delta S\), where \(\Delta H\) is the change in enthalpy and \(T\Delta S\) is the temperature times the change in entropy. With no heat exchange and no work done, \(\Delta H = 0\) and \(\Delta S = 0\). Therefore, \(\Delta G = 0\).

Key Concepts

Thermodynamics

Thermodynamics is a branch of physics that deals with heat and temperature, along with their relation to energy, work, radiation, and properties of matter. The behavior of these quantities is governed by the four laws of thermodynamics which dictate how systems at the macroscopic level, such as a gas sample, respond to changes in their surroundings.

For instance, when a gas expands, as described in the textbook exercise, thermodynamics helps us predict whether energy will be absorbed from the environment or released into it, and what changes might occur within the gas. Understanding thermodynamics is crucial for interpreting and predicting the outcome of thermal processes in scientific, industrial, and even everyday contexts.

Free Energy

Free energy, commonly symbolized as \( G \), represents the amount of work a system can perform when temperature and pressure are uniform through the system and the surroundings. It combines the system's internal energy with its entropy, a measure of disorder, to determine the energy that is available for doing useful work. For example, when we talk about a gas expanding with no change in temperature or external work as in our exercise, free energy can provide insight into the spontaneous nature of the process.

In thermodynamic processes, we often want to know if a reaction can happen spontaneously. The change in free energy, denoted as \( \Delta G \), helps make this determination. If \( \Delta G \)<0, the process is spontaneous, if \( \Delta G \)>0, the process is non-spontaneous, and if \( \Delta G \)=0, the system is in equilibrium and no net change is happening.

First Law of Thermodynamics

The first law of thermodynamics, also known as the law of energy conservation, states that energy cannot be created or destroyed in an isolated system. The total energy of the system is constant unless it is influenced by external work or heat transfer. This law is mathematically expressed as \( \Delta U = Q - W \), where \( \Delta U \) is the change in the internal energy of the system, \( Q \) is the heat added to the system, and \( W \) is the work done by the system. During the free expansion of an ideal gas, as in our exercise scenario, no work is done on or by the gas and no heat is transferred, resulting in zero change to the internal energy \( (\Delta U = 0)\).

Ideal Gas

An ideal gas is a hypothetical gas whose molecules occupy negligible space and have no interactions, allowing them to move randomly and independently of each other. The behavior of an ideal gas is described by the ideal gas law, \( PV=nRT \), where \( P \) is the pressure, \( V \) is the volume, \( n \) is the amount of gas (in moles), \( R \) is the ideal gas constant, and \( T \) is the temperature in Kelvin. Ideal gas models simplify the study of gases by focusing on their macroscopic properties without getting into the complexity of intermolecular forces.

In practical terms, an ideal gas allows us to predict the behavior of real gases under many conditions, as long as we acknowledge that deviations can occur under conditions of high pressure or low temperature where interactions and volume become significant.

Entropy

Entropy is a measure of the disorder or randomness in a system. In thermodynamics, it has a precise statistical definition based on the number of different microscopic states a system can have. Moreover, entropy is commonly associated with the second law of thermodynamics, which states that in an isolated system, entropy tends to increase over time.

In terms of thermodynamic processes, when a gas expands freely into a vacuum as seen in the exercise, its entropy increases because there are more possible positions and velocities for each molecule. However, in the case of an ideal gas that expands without changing its temperature or exchanging heat with the surroundings, which implies no change in its internal energy, the change in entropy \( (\Delta S) \) can still be zero, suggesting no additional disorder has been introduced to the system.

Enthalpy

Enthalpy, denoted by \( H \) , is a measurement of the total heat content of a system, often used in the context of chemical reactions and phase changes. It encompasses the internal energy of a system plus the product of its pressure and volume \( (H = U + PV) \).

Enthalpy becomes particularly useful for processes occurring at constant pressure, where the change in enthalpy \( (\Delta H) \) corresponds to the heat absorbed or released. However, in the case of our exercise, where a monoatomic ideal gas expands in a vacuum without changing temperature or doing external work, the change in enthalpy is zero \( (\Delta H = 0) \), indicating no heat exchange and no energy change related to pressure-volume work.