\( PV^n = k \) for process B and \( PV = k \) for process A.
\( PV = k \) for process B and A.
\( P^{n-1} = k \) for process B and \( T^n = k \) for process A.
\( T^n P^{n-1} = k \) for process A and \( PV = k \) for process B.
Hide Solution
Verified By Collegedunia
The Correct Option isA, C
Approach Solution - 1
The problem asks us to identify the correct thermodynamic equations that describe the processes A and B shown in the given Pressure-Volume (P-V) diagram.
Concept Used:
The key to solving this problem is to distinguish between different thermodynamic processes based on the slope of their curves on a P-V diagram.
Isothermal Process: A process that occurs at a constant temperature (\(T = \text{constant}\)). For an ideal gas, the equation of state is \(PV = nRT\), which simplifies to \(PV = \text{constant}\). The P-V curve is a hyperbola.
Adiabatic Process: A process that occurs with no heat exchange with the surroundings (\(Q = 0\)). The equation for an adiabatic process is \(PV^\gamma = \text{constant}\), where \( \gamma = C_p/C_v \) is the adiabatic index and is always greater than 1 (\(\gamma > 1\)).
Slope of P-V Curves: The slope of the P-V curve is given by \( \frac{dP}{dV} \).
For an isothermal process, \( P = k/V \), so the slope is \( \frac{dP}{dV} = -\frac{P}{V} \).
For an adiabatic process, \( P = k/V^\gamma \), so the slope is \( \frac{dP}{dV} = -\gamma \frac{P}{V} \).
Alternative forms of Adiabatic Equation: Using the ideal gas law, \(V = nRT/P\), the adiabatic equation \(PV^\gamma = k\) can be rewritten in terms of P and T: \[ P(nRT/P)^\gamma = k \implies P^{1-\gamma}T^\gamma = \text{constant} \implies \frac{T^\gamma}{P^{\gamma-1}} = \text{constant} \implies \frac{P^{\gamma-1}}{T^\gamma} = \text{constant} \]
Step-by-Step Solution:
Step 1: Analyze the slopes of the curves A and B in the P-V diagram.
The diagram shows two expansion processes (volume is increasing). We can visually inspect the steepness of the two curves. At any point of intersection with a vertical line (constant volume), curve B is steeper than curve A. Based on the concept that the adiabatic curve is steeper than the isothermal curve, we can identify the processes.
Process A: The curve is less steep. This corresponds to an isothermal process. Process B: The curve is steeper. This corresponds to an adiabatic process.
Step 2: Evaluate the given options based on this identification.
Option (1): \( pV^\gamma = k \) for process B and \( PV = k \) for process A.
For process B (adiabatic), the equation is indeed \(PV^\gamma = k\).
For process A (isothermal), the equation is indeed \(PV = k\).
This statement is correct.
Option (2): \( PV = k \) for process B and A.
This suggests both processes are isothermal, which is incorrect as they have different slopes. This statement is incorrect.
Option (3): \( \frac{P^{\gamma-1}}{T^\gamma} = k \) for process B and \( T = k \) for process A.
For process A (isothermal), the temperature is constant, so \(T = k\) is a correct description.
For process B (adiabatic), we need to check if \( \frac{P^{\gamma-1}}{T^\gamma} = k \) is a valid form of the adiabatic equation. As shown in the 'Concept Used' section, the relation \( P^{1-\gamma}T^\gamma = \text{constant} \) is correct for an adiabatic process. Rearranging this gives \( T^\gamma / P^{\gamma-1} = \text{constant} \), which means \( P^{\gamma-1} / T^\gamma \) is also a constant.
This statement is also correct.
Option (4): \( \frac{T^\gamma}{P^{\gamma-1}} = k \) for process A and \( pV = k \) for process B.
This incorrectly assigns the adiabatic relation to process A and the isothermal relation to process B. This statement is incorrect.
Final Computation & Result:
Based on our analysis, both Option (1) and Option (3) provide correct descriptions for the processes shown in the figure.
Therefore, the correct statements are given in (1) and (3).
Was this answer helpful?
0
0
Hide Solution
Verified By Collegedunia
Approach Solution -2
To determine the correct statements about the processes represented in the graph, we need to analyze the nature of processes A and B.
Process A is typically an isothermal process, which means that the temperature remains constant (\( T = \text{const} \)).
For an isothermal process, the equation \( PV = nRT \) holds true, indicating that the product of pressure and volume is constant.
Process B appears to be adiabatic since the slope is steeper than that of an isothermal process, which indicates that heat is not exchanged with the environment.
The adiabatic process is described by the equation \( PV^\gamma = \text{const} \), where \( \gamma = \frac{C_p}{C_v} \) is the ratio of specific heats.
Statements Analysis:
Statement (1): \( PV^n = k \) for process B is correct if process B follows an adiabatic path. \( PV = k \) for process A is also valid as it suggests constant temperature.
Statement (2): This statement does not hold true based on the analysis of processes A and B.
Statement (3): \( P^{n-1} = k \) for process B is valid for an adiabatic process, while \( T^n = k \) is a general representation for process A under isothermal conditions.
Statement (4): This statement is misleading; the relationship does not generally hold in the given form.
Thus, the correct statements regarding processes A and B are:
