Question

A reversible polytropic process is given by
1. \(\frac{{{T_1}}}{{{T_2}}} = {\left\{ {\frac{{{\rho _1}}}{{{\rho _2}}}} \right\}^{n}}\)
2. \(\frac{{{P_1}}}{{{P_2}}} = {\left\{ {\frac{{{\rho _1}}}{{{\rho _2}}}} \right\}^n}\)
3. \(\frac{{{T_1}}}{{{T_2}}} = {\left\{ {\frac{{{P_1}}}{{{P_2}}}} \right\}^{n - 1}}\)
4. \(\frac{{{T_1}}}{{{T_2}}} = {\left\{ {\frac{{{\rho _1}}}{{{\rho _2}}}} \right\}^{\frac{{n - 1}}{n}}}\)

Answer 1

Correct Answer - Option 2 : \(\frac{{{P_1}}}{{{P_2}}} = {\left\{ {\frac{{{\rho _1}}}{{{\rho _2}}}} \right\}^n}\)

Explanation:

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 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 Pvⁿ = 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: 
\(P{v^n} = C \Rightarrow \frac{P}{{{\rho ^n}}} = C \Rightarrow \frac{{{P_1}}}{{{P_2}}} = {\left( {\frac{{{\rho _1}}}{{{\rho _2}}}} \right)^n}\)

Value of n	Equation	Process
0	P = C	Isobaric
1	Pv = C	Isothermal
n	Pvn = C	Polytropic
γ (1.4)	Pvγ= C	Adiabatic
∞	v = C	Isochoric