Problem 23 - Chemistry The Central Science

Question

(a) How can we calculate \(\Delta S\) for an isothermal process? (b) Does \(\Delta S\) for a process depend on the path taken from the initial state to the final state of the system? Explain.

Step-by-Step Solution

Answer

(a) For an ideal gas in an isothermal process, we can calculate the change in entropy (ΔS) using the formula: ΔS = \(nR \ln\frac{V_f}{V_i}\) (b) ΔS for a process does not depend on the path taken from the initial state to the final state of the system, as entropy is a state function and only depends on the initial and final states (specifically, their volumes in this case).

Step 1: Define Entropy and Its Relation to Heat and Temperature

Entropy (S) is essentially a measure of the disorder or randomness of a system. For a given heat transfer (Q) at a constant temperature (T), the change in entropy (ΔS) can be defined as: ΔS = \(\frac{Q_{rev}}{T}\) where Q_rev is the heat transferred in a reversible process.

Step 2: Derive the Formula for ΔS in an Isothermal Process

In an isothermal process, the temperature (T) remains constant. For an ideal gas undergoing an isothermal reversible process, the heat Q_rev is given by: \(Q_{rev} = nRT \ln\frac{V_f}{V_i}\) where n is the number of moles of the ideal gas, R is the ideal gas constant, V_f is the final volume, and V_i is the initial volume. Now, we can use this relationship to determine the change in entropy for an isothermal process. By substituting Q_rev from the above equation into the formula for ΔS, we get: ΔS = \(\frac{nRT \ln\frac{V_f}{V_i}}{T}\) Upon simplification, we obtain: ΔS = \(nR \ln\frac{V_f}{V_i}\) This is the formula for the change in entropy for an ideal gas undergoing an isothermal process.

Step 3: Determine if ΔS Depends on the Process Path

Entropy is a state function, which means it depends only on the initial and final states of the system, not on the path taken during the process. For a given initial state and final state, the change in entropy (ΔS) will be the same regardless of the path taken from the initial to the final state. This is because the entropy change for an isothermal process can be expressed as a function of the initial and final volumes (V_i and V_f), as shown in the previous step: ΔS = \(nR \ln\frac{V_f}{V_i}\) Since this equation involves only the initial and final states (V_i and V_f) and not the specific path taken, ΔS does not depend on the process path. In conclusion: (a) To calculate the change in entropy (ΔS) for an ideal gas in an isothermal process, use the formula: ΔS = \(nR \ln\frac{V_f}{V_i}\) (b) The ΔS for a process does not depend on the path taken from the initial state to the final state of the system because entropy is a state function and depends only on the initial and final states (specifically, their volumes in this case).

Key Concepts

Isothermal Process

An isothermal process plays a crucial role in understanding thermodynamics and entropy change. During an isothermal process, a system undergoes a transformation while maintaining a constant temperature, which means that, despite the heat exchange occurring within the system, its internal temperature remains unchanged. This can occur naturally, in systems with very efficient heat exchange with their surroundings, or in a controlled laboratory setting.

In the context of entropy calculation, the constancy of temperature simplifies the process. Since temperature is steady, any heat transfer between the system and its environment during a reversible process can be directly related to the change in entropy without worrying about changing temperature variables. An ideal example could involve an ideal gas expanding or contracting in a sealed piston container with a perfect thermal reservoir around it, where no temperature variation occurs.

By understanding the behavior of processes under these specific conditions, students can grasp more complex thermodynamic concepts. It's also foundational for various real-world applications, such as improving the efficiency of heat engines and refrigerators.

State Function

When discussing entropy, it's fundamental to decipher the concept of a 'state function'. A state function is a property of a system that depends only on the current state of the system, not on the path by which the system arrived at that state. Examples of state functions include internal energy, enthalpy, and of course, entropy.

Envisioning a mountain hike can offer a simple analogy for a state function. The altitude of a hiker at the top of the mountain will be the same regardless of the path taken to get there. Similarly, the entropy of a thermodynamic system is like the altitude - it is determined solely by the system’s initial and final states. This inherent property simplifies entropy calculations because it eliminates the need to consider the myriad roads or transformations the system might experience from initial to final states.

Recognizing entropy as a state function aids students in understanding how it is used within the second law of thermodynamics, predicting whether processes are spontaneous, and in calculating the entropy changes for various thermodynamic processes.

Entropy Calculation

Entropy calculation involves determining the change in entropy, \(\Delta S\), during a thermodynamic process. It’s fundamental to recognize that entropy measures the disorder or randomness within a system. In the context of an isothermal process for an ideal gas, the calculation simplifies due to the temperature's constancy. Using the relationship \(\Delta S = \frac{Q_{rev}}{T}\), where \(Q_{rev}\) is the heat involved in a reversible process and \(T\) is the constant temperature, the entropy change can be calculated with ease.

For ideal gases, this relationship can be articulated more precisely while considering the change in volume of the gas. Here, the heat transfer \(Q_{rev}\) is linked to the number of moles \(n\), the ideal gas constant \(R\), and the logarithm of the final volume to the initial volume ratio, leading to the formula \(\Delta S = nR \ln\frac{V_f}{V_i}\).

This formula illuminates the direct connection between entropy change and volume change in an isothermal process. The calculation is therefore highly relevant in situations involving heat engines, where the control of volume changes is synonymous with energy efficiency.