Question

When an ideal gas at pressure $P$, temperature $T$ and volume $V$ is isothermally compressed to $\dfrac{v}{n}$, its pressure becomes ${{P}_{i}}$. If the gas is compressed adiabatically to $\dfrac{V}{n}$, its pressure becomes${{P}_{a}}$. The ratio of $\dfrac{{{P}_{i}}}{{{P}_{a}}}$ is $\left( \gamma =\dfrac{{{C}_{P}}}{{{C}_{v}}} \right)$
$\begin{align}
  & A.1 \\
 & B.n \\
 & C.{{n}^{\gamma }} \\
 & D.{{n}^{1-\gamma }} \\
\end{align}$

Answer 1

Hint: The basic equation of isothermal and adiabatic process helps us to get into an answer for this question. The equation for isothermal process is
$PV=c$
Where c is a constant.
And the equation for adiabatic process is $P{{V}^{\gamma }}=c$
$P{{V}^{\gamma }}=c$
Where c is a constant. First apply the given condition in the equation of isothermal process. Then apply the adiabatic equation also. Find the pressure from both of these and take the ratio.

Complete step by step answer:
First of all let us take a look at what a process of isothermal and adiabatic is. An isothermal process is a process in thermodynamics in which the temperature of a system remains constant. The transfer of heat into or out of the system happens so slowly in order to maintain thermal equilibrium. Condensation is an example of this process. All the reactions going on in the refrigerator is an isothermal because a constant temperature is maintained in it. An adiabatic process is a thermodynamic process that occurs without transfer of heat or mass between a thermodynamic system and its surroundings. Opposite to that of an isothermal process, an adiabatic process transfers energy into the surroundings only in the form of work.
In this question, for an isothermal process,
$PV=c$
Where c is constant.

Therefore
$PV={{P}_{i}}\dfrac{V}{n}$${{P}_{a}}{{\left( \dfrac{V}{n} \right)}^{\gamma }}=P{{V}^{\gamma }}$
Hence we can write that
${{P}_{i}}=nP$………. (1)
And in an adiabatic process,
$P{{V}^{\gamma }}=c$
Where c is a constant.
Therefore,
${{P}_{a}}{{\left( {{V}_{a}} \right)}^{\gamma }}=P{{V}^{\gamma }}$
Else we can write that,
${{P}_{a}}{{\left( \dfrac{V}{n} \right)}^{\gamma }}=P{{V}^{\gamma }}$
Or
${{P}_{a}}={{n}^{\gamma }}P$………… (2)
Equating equation 1 and 2will give,
$\dfrac{{{P}_{i}}}{{{P}_{a}}}=\dfrac{n}{{{n}^{\gamma }}}={{n}^{\left( 1-\gamma \right)}}$

So, the correct answer is “Option D”.

Note: It is a thermodynamic process in which a gas compression and heat is being produced. One of the simple examples is the release of air from a pneumatic tire. Devices such as nozzles, compressors, and turbines are good applications of adiabatic processes.