Question

The pressure on \(0.850\) mol of neon gas is increased from \(1.25\) atm to \(2.75\) atm at \(100{ }^{\circ} \mathrm{C}\). Assuming the gas to be ideal, calculate \(\Delta S\) for this process.

Short answer

The change in entropy (ΔS) for the process can be calculated using the formula ΔS = n * R * ln(P2/P1), where n = 0.850 mol, R = 0.0821 L⋅atm/(mol⋅K), and P2/P1 = 2.75 atm / 1.25 atm = 2.2. Plugging these values into the formula gives ΔS ≈ 0.0552 L⋅atm/K.

Step-by-step solution

Identify the given values and formula to use

We are given the following values: - Number of moles (n) = 0.850 mol - Initial pressure (P1) = 1.25 atm - Final pressure (P2) = 2.75 atm - Temperature (T) = 100 °C = 373.15 K (converting to Kelvin) The formula to find the change in entropy for an ideal gas under constant temperature is: ΔS = n * R * ln(P2/P1) Where ΔS is the change in entropy, n is the number of moles, R is the gas constant (8.314 J/(mol⋅K) for SI units but for this problem, we will use R = 0.0821 L⋅atm/(mol⋅K)) and ln is the natural logarithm. Our main task is to find ΔS using the given information.

Calculating the pressure ratio

First, let's find the pressure ratio, which is P2/P1. Divide the final pressure (2.75 atm) by the initial pressure (1.25 atm) to get the pressure ratio: Pressure ratio = P2/P1 = 2.75 atm / 1.25 atm

Applying the formula for ΔS

Now, we apply the formula for the change in entropy (ΔS) using the known values: ΔS = n * R * ln(P2/P1) ΔS = (0.850 mol) * (0.0821 L⋅atm/(mol⋅K)) * ln(2.75/1.25)

Solving for ΔS

Finally, we can calculate the change in entropy (ΔS) by plugging the values into the above equation: ΔS = (0.850 mol) * (0.0821 L⋅atm/(mol⋅K)) * ln(2.2) ΔS ≈ 0.850 * 0.0821 * 0.7885 ΔS ≈ 0.0552 L⋅atm/K The change in entropy (ΔS) for this process is approximately 0.0552 L⋅atm/K.

Key concepts

Ideal Gas Law

The Ideal Gas Law is a fundamental equation in the study of thermodynamics which is pivotal for understanding how gases behave under various conditions of temperature, pressure, and volume. It is expressed as PV=nRT, where P represents pressure, V is volume, n is the number of moles of gas, R is the gas constant, and T is the absolute temperature in Kelvin.

When we refer to an 'ideal' gas, we are talking about a hypothetical gas that perfectly follows this law without any deviations, which real gases do at high pressures or low temperatures. The Ideal Gas Law allows us to predict the behavior of real gases in many common situations and is the basis for deriving other important formulas in thermodynamics, including those related to entropy.

For instance, in the textbook exercise, the Ideal Gas Law assists in understanding how the pressure change at a constant temperature impacts the gas's entropy. The equation emphasizes the proportional relationship between pressure and volume (when temperature is constant), which lays the groundwork for understanding how entropy, a measure of disorder or randomness in a system, changes with pressure.

Thermodynamics

Thermodynamics is an essential branch of physics concerned with heat and temperature and their relation to energy and work. It involves studying and understanding the conversion of energy from one form to another, and how energy affects matter. There are four laws of thermodynamics which govern the principles of energy transfer and entropy.

The first law, also known as the law of conservation of energy, asserts that energy cannot be created or destroyed, only transformed. The second law is the one most pertinent to the exercise, stating that entropy of an isolated system always increases over time. Essentially, it implies that natural processes have a preferred direction - from less to more disordered states.

In the context of the problem, thermodynamics provides the framework for determining how the entropy of a system changes. By focusing on the second law, we can predict that compressing the gas (increasing its pressure) while maintaining a constant temperature leads to a change in the system’s entropy, although the total energy within the system remains constant.

Entropy

Entropy is a measure of disorder or randomness in a system, a key concept in the second law of thermodynamics. It's also considered a measure of the number of microscopic configurations that correspond to a thermodynamic system's macroscopic state.

Entropy can also be viewed as a measure of unpredictability or the dissemination of energy: the higher the entropy, the greater the disorder and the less energy available to do work. In practical terms, entropy quantifies the energy in a system that is no longer capable of performing mechanical work. As entropy increases, systems tend naturally to go from a state of order (low entropy) to disorder (high entropy).

In calculations, such as the one presented in the exercise, entropy is mathematically denoted by the symbol ΔS. Calculating changes in entropy, like the problem at hand, is critical in understanding the direction of energy flow and the feasibility of chemical and physical processes. It helps us predict how changes in heat, work, temperature, or matter could affect the overall disorder of the system.