One mole of an ideal gas undergoes two different cyclic processes I and I, as shown in the P-V diagrams below. In cycle, I processes a, b, c and d are isobaric, isothermal, isobaric, and isochoric, respectively. In cycle II, processes a',b',c', and d' are isothermal, isochoric, isobaric, and isochoric, respectively. The total work done during cycle I is πI and that during cycle II is πII. The ratio WI/WII is
Correct Answer: 2
Approach Solution - 1
Thermodynamic Cycles and Processes
We are considering an ideal gas that undergoes two distinct cyclic processes, denoted as I and II. The stages of the first cycle, Cycle I, are represented as follows:
Cycle I: Stages a, b, c, and d
- Stage a: Isobaric transformation (constant pressure)
- Stage b: Isothermal transformation (constant temperature)
- Stage c: Isobaric transformation (constant pressure)
- Stage d: Isochoric transformation (constant volume)
Explanation of Stages in Cycle I:
1. **Isobaric (Stage a and c)**: In this stage, the pressure of the gas remains constant while the volume changes. During an isobaric process, the work done on or by the gas is given by \( W = P \Delta V \), where \( P \) is the pressure, and \( \Delta V \) is the change in volume.
2. **Isothermal (Stage b)**: In this stage, the temperature of the gas remains constant. For an ideal gas undergoing an isothermal process, the internal energy change is zero, and the work done is given by the equation \( W = nRT \ln \left( \frac{V_f}{V_i} \right) \), where \( V_f \) and \( V_i \) are the final and initial volumes, respectively.
3. **Isochoric (Stage d)**: In this stage, the volume of the gas remains constant. Since there is no change in volume, no work is done during the isochoric process. However, heat may be transferred, resulting in a change in the internal energy of the gas.
Conclusion
As described, the stages of cycle I include isobaric, isothermal, isobaric, and isochoric transformations. The answer to the problem, based on this analysis, is:
The answer is 2.
Approach Solution -2
\[ W_I = W_a + W_b + W_c + W_d \]\[ = 4P_0 \left(2V_0 - V_0\right) + nRT \ln\left(\frac{4V_0}{2V_0}\right) + 2P_0 \left(V_0 - 4V_0\right) + 0 \] \[ = 4P_0 V_0 + nRT \ln\left(\frac{8P_0 V_0}{nR}\right) \ln 2 - 6P_0 V_0 \] \[ = 8P_0 V_0 \ln 2 - 2P_0 V_0 \] \[ W_{II} = W_a' + W_b' + W_c' + W_d' \] \[ = nRT \ln\left(\frac{2V_0}{V_0}\right) + 0 + P_0 \left(V_0 - 2V_0\right) + 0 \] \[ = nRT \ln\left(\frac{4P_0 V_0}{nR}\right) \ln 2 - P_0 V_0 \] \[ = 4P_0 V_0 \ln 2 - P_0 V_0 \] \[ \frac{W_I}{W_{II}} = \frac{8P_0 V_0 \ln 2 - 2P_0 V_0}{4P_0 V_0 \ln 2 - P_0 V_0} = 2 \] The ratio \( \frac{W_I}{W_{II}} \) is 2,Therefore the Correct Answer is 2.
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Concepts Used:
Thermodynamics
Thermodynamics in physics is a branch that deals with heat, work and temperature, and their relation to energy, radiation and physical properties of matter.
Important Terms
System
A thermodynamic system is a specific portion of matter with a definite boundary on which our attention is focused. The system boundary may be real or imaginary, fixed or deformable.
There are three types of systems:
- Isolated System β An isolated system cannot exchange both energy and mass with its surroundings. The universe is considered an isolated system.
- Closed System β Across the boundary of the closed system, the transfer of energy takes place but the transfer of mass doesnβt take place. Refrigerators and compression of gas in the piston-cylinder assembly are examples of closed systems.
- Open System β In an open system, the mass and energy both may be transferred between the system and surroundings. A steam turbine is an example of an open system.
Thermodynamic Process
A system undergoes a thermodynamic process when there is some energetic change within the system that is associated with changes in pressure, volume and internal energy.
There are four types of thermodynamic process that have their unique properties, and they are:
- Adiabatic Process β A process in which no heat transfer takes place.
- Isochoric Process β A thermodynamic process taking place at constant volume is known as the isochoric process.
- Isobaric Process β A process in which no change in pressure occurs.
- Isothermal Process β A process in which no change in temperature occurs.
Laws of Thermodynamics
Zeroth Law of Thermodynamics
The Zeroth law of thermodynamics states that if two bodies are individually in equilibrium with a separate third body, then the first two bodies are also in thermal equilibrium with each other.
First Law of Thermodynamics
The First law of thermodynamics is a version of the law of conservation of energy, adapted for thermodynamic processes, distinguishing three kinds of transfer of energy, as heat, as thermodynamic work, and as energy associated with matter transfer, and relating them to a function of a body's state, called internal energy.
Second Law of Thermodynamics
The Second law of thermodynamics is a physical law of thermodynamics about heat and loss in its conversion.
Third Law of Thermodynamics
Third law of thermodynamics states, regarding the properties of closed systems in thermodynamic equilibrium: The entropy of a system approaches a constant value when its temperature approaches absolute zero.