In the polytropic process, equation pvn = constant, if n = ∞, the process is called
(a) constant volume (d) adiabatic
(b) constant pressure (e) isothermal
(c) constant temperature
The correct answer and explanation is :
In a polytropic process, the general equation is given by:
[
p v^n = \text{constant}
]
where:
- ( p ) is the pressure,
- ( v ) is the volume,
- ( n ) is the polytropic index.
For different values of ( n ), the process behaves in different ways. Let’s break down the case when ( n ) approaches infinity (( n = \infty )):
Understanding ( n = \infty ):
When ( n = \infty ), the equation becomes:
[
p v^\infty = \text{constant}
]
In this scenario, for the equation to hold true, the volume ( v ) must remain constant because raising any finite value of volume to an infinitely large power will tend toward zero unless the volume is kept constant. If the volume is constant, then the pressure must also remain constant in order for the equation ( p v^\infty = \text{constant} ) to hold. This implies that the process is happening at constant volume.
Why constant volume?
If the volume doesn’t change during the process, then any change in pressure would correspond to a change in temperature for an ideal gas (according to the Ideal Gas Law, ( pV = nRT )). In a real-world application, this could mean a rigid container where the volume is fixed and the pressure could change.
Conclusion:
Thus, when ( n = \infty ), the process is called constant volume, because the volume does not change, and the pressure is the only variable that can adjust.
Correct Answer: (a) constant volume
Explanation:
- Constant Volume: At ( n = \infty ), the equation suggests that volume remains unchanged throughout the process.
- Constant Pressure (b): This would happen at ( n = 0 ), not ( \infty ).
- Adiabatic (d): An adiabatic process occurs when no heat is exchanged with the surroundings, which corresponds to ( n = \gamma ), the ratio of specific heats, not ( n = \infty ).
- Isothermal (e): In an isothermal process, temperature is constant, and this corresponds to ( n = 1 ), not ( n = \infty ).
So, the correct answer is ( \boxed{\text{(a) constant volume}} ).
