An ideal gas undergoes an isothermal expansion at \(77^{\circ} \mathrm{C}\) increasing its volume from \(1.3\) to \(3.4 \mathrm{~L}\). The entropy change of the gas is \(24 \mathrm{~J} / \mathrm{K}\). How many moles of gas are present?

The ideal gas law is a pivotal concept in understanding the behavior of gases under various conditions. It is an equation of state for a hypothetical gas called an 'ideal gas,' which is a simplified model where the molecules do not attract or repel each other and always occupy non-zero volume. The law is usually stated as:

\( PV = nRT \)
where:\

• \(P\) stands for pressure,
• \(V\) is the volume,
• \(n\) represents the number of moles,
• \(R\) is the gas constant, and
• \(T\) is the temperature in Kelvin.

This equation shows that, for a given amount of gas, the pressure and volume are directly proportional to the temperature. It's essential to note that the temperature needs to be in Kelvin for the equation to be accurate. In this isothermal expansion problem, even though the temperature is constant, the ideal gas law still governs the relationship between the pressure, volume, and number of moles of the gas.

Entropy, symbolized by \( S \), is a measure of the disorder, or randomness, of the particles that make up a system. In thermodynamics, it's a key indicator of the energy distribution within a system at a certain temperature. When a gas expands isothermally, it undergoes a change in entropy because the gas particles occupy a more extensive volume and hence have more possible positions and momenta.

The entropy change for an isothermal process undergone by an ideal gas can be calculated using the formula:

\( \triangle S = nR\text{ln}\left(\frac{V_2}{V_1}\right) \)
where \( n \) is the number of moles of gas, \( R \) is the gas constant, and \( V_1 \) and \( V_2 \) are the initial and final volumes respectively. The natural logarithm \( (ln) \) is used because of the nature of the expansion. A positive change in entropy indicates an increase in disorder and vice-versa for a negative change.

The mole is a fundamental unit in chemistry for measuring an amount of substance. One mole of any substance contains Avogadro's number of molecules, which is approximately \(6.022 \times 10^{23}\). In thermodynamics and gas law problems, the moles calculation is essential as it often relates to the volume, pressure, and temperature of the gas.

To calculate the moles of gas in the given problem, we start with the entropy change equation:

\( n = \frac{\triangle S}{R\text{ln}\left(\frac{V_2}{V_1}\right)} \)
All variables except for the number of moles \( (n) \) are known. By inserting the entropy change, the gas constant, and the volumes into the equation, the moles of gas can be directly calculated. Moles calculation is crucial as it allows scientists and engineers to predict and manipulate the behavior of gases in various chemical and physical processes.

Thermodynamics is the branch of physics that deals with the relationships between heat and other forms of energy. In essence, it looks at how thermal energy is transformed into other energy forms and how it affects matter. The laws of thermodynamics govern the principles underlying heat transfer, energy conversions, and the physical and chemical property changes of substances.

An isothermal expansion, as described in the exercise, is a process that occurs at a constant temperature, meaning thermal energy is either absorbed or released to maintain the same temperature. Such processes are a key study area in thermodynamics because they illustrate how gases behave under thermal equilibrium and provide insights into the conservation of energy within a system. Understanding these concepts helps in analyzing engines, refrigerators, and various other systems where energy conversion and gas properties are of interest.