Question

(a) What sign for $\mathrm{\Delta }S$ do you expect when the volume of $0.200$ mol of an ideal gas at ${27}^{\circ }\mathrm{C}$ is increased isothermally from an initial volume of $10.0\mathrm{L}$ ? (b) If the final volume is $18.5\mathrm{L}$ calculate the entropy change for the process. (c) Do you need to specify the temperature to calculate the entropy change? Explain.

Short answer

The sign of entropy change when the volume of an ideal gas is increased isothermally is positive, indicating an increase in system entropy. The entropy change for the given process is approximately 2.234 J/K. Specifying the temperature isn't directly necessary for calculating the entropy change but indirectly affects it and characterizes the process as an isothermal expansion.

Step-by-step solution

Determine the sign of entropy change when the volume of an ideal gas is increased isothermally

Since the volume of the ideal gas is increased in an isothermal process, it means that more microstates are available for the gas particles due to an increase in volume while keeping the temperature constant. Higher number of microstates corresponds to greater disorder, which ultimately leads to an increase in entropy. Hence, we expect the sign of entropy change, ∆S, to be positive.

Calculate the entropy change for the process

To calculate the entropy change, we will use the formula for entropy change of an ideal gas in an isothermal process: ∆S = nRln(Vf/Vi) Where: n = number of moles R = gas constant = 8.314 J/(mol K) Vf = final volume Vi = initial volume Given values: n = 0.200 mol Temperature = 27°C = 27 + 273.15 = 300.15 K Initial volume, Vi = 10.0 L = 10.0 x 10^(-3) m³ Final volume, Vf = 18.5 L = 18.5 x 10^(-3) m³ Now we can plug these values into the entropy change formula: ∆S = 0.200 * 8.314 * ln(18.5 x 10^(-3) / 10.0 x 10^(-3)) ∆S ≈ 2.234 J/K

Determine if specifying temperature is necessary to calculate entropy change

The temperature is essential to understand the characteristic of the process, i.e., isothermal expansion. But for the given problem, the temperature does not explicitly appear in the entropy change formula. However, temperature plays an indirect role: the isothermal process means that we hold the temperature constant throughout the process, which affects the entropy change. So, the answer to this question is: for this specific problem, specifying the temperature was not necessary to calculate the entropy change, but it helped characterize the process as an isothermal expansion and indirectly influenced the entropy change. #Summary# The sign of entropy change for the isothermal expansion of an ideal gas is positive, meaning the system entropy increases. The entropy change for the given process is approximately 2.234 J/K. While specifying the temperature isn't directly necessary for calculating the entropy change, it indirectly affects the entropy change and helps characterize the process as an isothermal expansion.

Key concepts

Isothermal Expansion

Isothermal expansion occurs when a gas expands at a constant temperature. This means that while the gas is allowed to expand, the temperature remains unchanged. In such processes, energy is used to perform work on the surroundings instead of changing the internal energy (since temperature change is not involved).
For instance, if you have a gas in a piston, and the piston moves outwards while keeping the temperature constant, this is isothermal expansion. This is different from adiabatic expansion, where no heat is exchanged with surroundings, leading to a temperature change. In isothermal expansion, the work done by the gas results in an increase in its volume while maintaining constant thermal conditions. This process is common in thermodynamic applications, where maintaining temperature stability is essential.

Ideal Gas Law

The Ideal Gas Law is a fundamental equation in thermodynamics, represented by $PV=nRT$. It connects pressure $P$, volume $V$, and temperature $T$ of a given amount of ideal gas with the number of moles $n$ and the ideal gas constant $R$. This law provides a great approximation for the behavior of gases under many conditions.

• $P$ stands for the pressure exerted by the gas.
• $V$ represents the volume that the gas occupies.
• $n$ is the number of moles of the gas.
• $R$ is known as the ideal gas constant.
• $T$ is the temperature of the gas in Kelvin.

This law is pivotal in predicting how a gas will react to changes in its environment like changes in pressure or temperature. Though it assumes ideal conditions, deviations can occur at high pressures or low temperatures when gases behave more like real gases.

Gas Constant

The gas constant, symbolized as $R$, is a proportionality factor in the Ideal Gas Law. Its value is approximately $8.314$ J/(mol K). This constant is derived from experiments and is essential in calculations involving gases. It helps relate energy (Joules) to the thermal and physical properties (molar) of gases.
While $R$ maintains a constant value across calculations, the significance of $R$ lies in its ability to integrate various units in gas equations. Its consistent presence ensures accuracy and standardization in measuring gas behavior. For students dealing with gas-related problems, recognizing the use and importance of $R$ will aid in understanding how different aspects of gas laws interconnect.

Microstates

In thermodynamics, microstates pertain to the different configurations that a system can achieve at a molecular level, given the same macroscopic conditions. The concept of microstates is fundamental when understanding entropy, as entropy is associated with the number of microstates of a system. More microstates mean more ways to arrange the system without altering its macroscopic properties, leading to greater disorder.
For example, during an isothermal expansion, as a gas expands, the number of microstates increases because molecules have more space to occupy. This leads to an increase in the disorder of the system, and thus, an increase in entropy. Recognizing the role of microstates is crucial as it links the microscopic phenomena to macroscopic observables like entropy and temperature.

Thermodynamics

Thermodynamics is a branch of physics that examines how heat interacts with physical systems. It delves into concepts of energy, entropy, and the laws governing the interactions among matter. These principles are essential in a wide array of practical applications, ranging from engines and refrigerators to the human body and even the universe itself.
Key laws underpin thermodynamics, explaining how energy is conserved and transferred.

• The First Law, or the Law of Energy Conservation, states that energy cannot be created or destroyed, only transformed.
• The Second Law introduces the concept of entropy, emphasizing that systems tend toward increasing disorder.
• The Third Law of Thermodynamics asserts that as temperature approaches absolute zero, the entropy of a perfect crystal approaches a constant minimal value.

Understanding these laws provides insight into how energy flows and transforms, helping not only in academic pursuit but also in understanding everyday phenomena.