Chapter 4: Problem 71
When one mole of an ideal gas is compressed to half of its initial volume and simultaneously heated to twice its initial temperature, the change in entropy of gas \((\Delta S)\) is : (a) \(C_{p, m} \ln 2\) (b) \(C_{v, m} \ln 2\) (c) \(R \ln 2\) (d) \(\left(C_{v, m}-R\right) \ln 2\)
Short Answer
Expert verified
\(\left(R - C_{v, m}\right) \ln 2\)
Step by step solution
01
Understand the process
The exercise describes a thermodynamic process in which one mole of an ideal gas is compressed to half its initial volume while its temperature is doubled. This implies an isentropic process and uses the ideal gas law, combined with the thermodynamic relation for entropy change.
02
Calculate entropy change
To find the change in entropy \(\Delta S\), use the formula for a reversible process: \[\Delta S = nC_{v, m} \ln\frac{V_2}{V_1} + nR\ln\frac{T_2}{T_1}\] Since the volume is halved, \(V_2/V_1 = 1/2\), and the temperature is doubled, \(T_2/T_1 = 2\), the formula simplifies to: \[\Delta S = nC_{v, m} \ln(1/2) + nR\ln(2)\]
03
Simplify the entropy change formula
Using the properties of logarithms, we can simplify the formula as: \[\Delta S = nC_{v, m} (\ln 1 - \ln 2) + nR(\ln 2 - \ln 1)\] Since \ln 1 = 0, the formula further simplifies to: \[\Delta S = -nC_{v, m} \ln 2 + nR \ln 2\] Given that \(n = 1\) mole, we combine terms to get the final result: \[\Delta S = (R - C_{v, m}) \ln 2\]
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Thermodynamic Processes
Thermodynamic processes describe the dynamics of systems in which energy in the form of heat or work is transferred to or from the surroundings. These include processes like heating, cooling, compressing, or expanding a substance.
Understanding thermodynamic processes is crucial for calculating changes in properties such as temperature, pressure, volume, and entropy. Entropy, a central concept in thermodynamics, quantifies the amount of disorder within a system. It's associated with the number of microscopic configurations that correspond to a thermodynamic system's macroscopic state.
An important aspect of thermodynamics is the laws that govern these processes. The first law relates to the conservation of energy, indicating that energy cannot be created or destroyed. The second law introduces the concept of entropy and suggests that for any spontaneous process, the total entropy of a system and its environment always increases.
The exercise example highlights a thermodynamic process involving entropy change and requires applying these principles to reach a solution.
Understanding thermodynamic processes is crucial for calculating changes in properties such as temperature, pressure, volume, and entropy. Entropy, a central concept in thermodynamics, quantifies the amount of disorder within a system. It's associated with the number of microscopic configurations that correspond to a thermodynamic system's macroscopic state.
An important aspect of thermodynamics is the laws that govern these processes. The first law relates to the conservation of energy, indicating that energy cannot be created or destroyed. The second law introduces the concept of entropy and suggests that for any spontaneous process, the total entropy of a system and its environment always increases.
Understanding Reversible and Irreversible Processes
Reversible processes are idealized scenarios where the system remains in equilibrium as it changes states, meaning the process can be reversed without any entropy change in the universe. In contrast, an irreversible process is one that increases the total entropy of the universe.The exercise example highlights a thermodynamic process involving entropy change and requires applying these principles to reach a solution.
Ideal Gas Law
The Ideal Gas Law is a fundamental equation that relates the pressure (P), volume (V), temperature (T), and amount of substance (n) in moles of an ideal gas. Expressed mathematically as:
\[ PV = nRT \]
The constant R is known as the ideal or universal gas constant, and the law serves as a good approximation for the behavior of real gases under many conditions, although it assumes that gas particles have no volume and do not interact.
Understanding the ideal gas law is critical when studying thermodynamic processes since it allows us to predict how a gas will respond to changes in conditions such as pressure, volume, or temperature. For instance, in the given exercise, knowing that the volume is compressed to half and the temperature is doubled provides essential information for calculating the change in entropy of the gas using this law.
\[ PV = nRT \]
The constant R is known as the ideal or universal gas constant, and the law serves as a good approximation for the behavior of real gases under many conditions, although it assumes that gas particles have no volume and do not interact.
Understanding the ideal gas law is critical when studying thermodynamic processes since it allows us to predict how a gas will respond to changes in conditions such as pressure, volume, or temperature. For instance, in the given exercise, knowing that the volume is compressed to half and the temperature is doubled provides essential information for calculating the change in entropy of the gas using this law.
Isentropic Process
An isentropic process is a thermodynamic process that occurs at a constant entropy, which means there is no change in the entropy of the system as the process unfolds. Isentropic processes are reversible adiabatic processes; they involve no heat transfer with the surroundings (hence adiabatic), and the total entropy remains unchanged (hence isentropic).
In an idealized isentropic process, entropy is conserved due to the absence of any external entropy exchange, and it's manifested in perfect energy conservation within the system. This implies that all changes are internally reversible and there are no frictional losses or non-conservative forces at play.
In an idealized isentropic process, entropy is conserved due to the absence of any external entropy exchange, and it's manifested in perfect energy conservation within the system. This implies that all changes are internally reversible and there are no frictional losses or non-conservative forces at play.
- Significance in Real-World Applications: Processes that approximate isentropic behavior are important in engineering, for example in the functioning of steam turbines or compressors, where they are used to maximize efficiency.
- Practical Example: The exercise provided is similar to an isentropic process, but since there is a change in entropy due to heating, it strictly doesn't qualify as one. Understanding where real-life deviates from the ideal helps in better understanding system behavior.