A process in which no heat is supplied and rejected is known as an adiabatic process. But if entropy is not constant, then this process is called the polytropic process.
Hint: This question can be solved by eliminating the options.
Isothermal Process:
An isothermal process is a process in which temperature remains constant. i.e. T_{1} = T_{2} ⇒ ΔU = 0
For the isothermal process δQ = -δW ≠ 0
Isentropic Process: ΔS = 0
Hyperbolic Process is the isothermal process
So the process cannot be isothermal/hyperbolic and isentropic. It can only be polytropic.
In many real processes, it is found that the states during an expansion or compression can be described approximately by a relation of the form Pvn = constant,
where n is constant called index of compression or expansion, p and v are the average value of pressure and specific volume for the gases.
The process Pvn = constant is called the polytropic process.
when this n value = 1; then the process is called Isothermal process and,
when this n value = γ; then the process is called an adiabatic process.
Hence Pvn = C is called the general law for the expansion or compression of gases.
Additional Information
PVn = C
n = 0 ⇒ P = C ⇒ Constant Pressure Process (Isobaric Process)
n = 1 ⇒ PV = C ⇒ Constant Temperature Process (Isothermal process)
n = γ ⇒ PVγ = C ⇒ Adiabatic Process
n = ∞ ⇒ V = C ⇒ Constant Volume Process (Isochoric process)
In many real processes, it is found that the states during an expansion or compression can be described approximately by a relation of the form Pv^{n} = constant,
where n is a constant called index of compression or expansion, P and v are the average value of pressure and specific volume for the system.
Compressions and expansions of the form Pv^{n} = constant are called polytropic process.
For the reversible polytropic process, single values of P and v can truly define the state of a system, dW = -Pdv.
The equation for the polytropic process: \(P{v^n} = C \Rightarrow \frac{P}{{{\rho ^n}}} = C \Rightarrow \frac{{{P_1}}}{{{P_2}}} = {\left( {\frac{{{\rho _1}}}{{{\rho _2}}}} \right)^n}\)
A piston-cylinder device with air at an initial temperature of 30°C undergoes an expansion process for which pressure and volume are related as given below:
p (kPa)
100
37.9
14.4
V (m^{3})
0.1
0.2
0.4
The work done by the system for n = 1.4 will be
4.8 kJ
6.8 kJ
8.4 kJ
10.6 kJ
Answer (Detailed Solution Below)
Option 4 : 10.6 kJ
Polytropic Process MCQ Question 7 Detailed Solution
For the polytropic process with n values ranging from 1 to γ, where γ is heat capacity ratio. Which of the following option supports the fact that specific heat capacity of such process is negative?
Heat supplied in the system > work done by the system
Heat supplied in the system < work done by the system
Heat supplied in the system = work done by the system
None of the above
Answer (Detailed Solution Below)
Option 2 : Heat supplied in the system < work done by the system
Polytropic Process MCQ Question 9 Detailed Solution
If in any process the temperature of the system decreases even after supplying the heat then the specific heat capacity of such process is considered as negative.
Now, in the above case:
If Q < W, then an extra amount of energy is required to complete the work, other than the energy supplied by heat (Q).
This extra amount of energy comes from the internal energy, hence the internal energy decreases.
Since internal energy (U) is the function of temperature only, so the temperature of the system will also decrease.
So in this process, we can see that even while we are supplying the heat the temperature is decreasing which indicates that the specific heat capacity of the process is negative.