Question

The temperature v/s entropy diagram is indicated in the figure. Its PV equivalent diagram will be given as,

A.
 

B.

C.

D.

Answer 1

Hint: For an isothermal process, the product of pressure and volume will be equal to the product of number of moles, universal gas constant and temperature. This will be equal to a constant as temperature is a constant. And for adiabatic processes, the product of pressure and the volume rise to adiabatic index will be constant.

Complete step by step answer:
When we notice the diagram, we can see that the process AB and DC are isothermal processes. Therefore we can write that the product of pressure and volume will be equal to the product of number of moles, universal gas constant and temperature. This will be equal to a constant as temperature is a constant. That is,
$PV=nRT=\text{constant}\left( \because T\text{ constant} \right)$
Where $P$ be the pressure, $V$be the volume, $n$ be the number of moles, $R$be the universal gas constant and $T$be the temperature.
Therefore we can say that the P-V plot for these two isothermal processes will be hyperbolas.
And when we check the diagram once more, we can see that the process AD and BC are having a fixed entropy. Therefore, they are termed as constant entropy processes. For such a situation we can say that the product of pressure and the volume rise to adiabatic index will be constant. That is,
$P{{V}^{\gamma }}=\text{constant}$
As we all know that the slope of constant entropy termed as adiabatic process is higher than the slope of an isothermal process.
Hence we can say that the ratio of the slope of the isothermal curve to the slope of adiabatic curve should always be greater than one.
$\dfrac{\text{slope of adiabatic curve}}{\text{slope of isothermal curve}}=\gamma >1$

So, the correct answer is “Option A”.

Note:
The adiabatic index is the ratio of the specific heats of at constant pressure to the specific heat at the constant volume. It is a factor for calculating the speed of sound in a gas and in other adiabatic processes and also in the calculations of heat engines.it is also called as isentropic expansion factor.