Question

When an ideal gas with pressure P and volume V is compressed isothermally to one fourth of its volume, the pressure is P1. When the same gas is compressed polytropically according to the equation PV1.5 = constant to one fourth of its initial volume, the pressure is P2. The ratio of (P1/P2) is

Answer 1

Answer:

The ratio of (P₁/P₂)  is equal to 1/2.

Explanation:

For Isothermal process, $PV=constant$

                            $PV=P_{1}( \frac{V}{4} )$

                            $P_{1} =4P$                          ...................(1)

For polytropically process,  $PV^{1.5} =constant$

                            $PV^{1.5} =P_{2}( \frac{V}{4})^{1.5}$

                            $P_{2}=8P$                        ....................(2)

By dividing eq. (1) by eq.(2)

                             $\frac{P_{1} }{P_{2} } =\frac{1}{2}$

Therefore,  the ratio of  $\frac{P_{1} }{P_{2} }$ is equal to  $\frac{1}{2}$.

Answer 2

Answer:

The ratio of (P₁/P₂)  is equal to $\frac{1}{2}$.

Explanation:

• For Isothermal process, we know that temperature remains constant
  $PV = nRT$
• Now as we know n and R are number of moles and Rydberg's constant hence
  $PV = Constant$
  $P_{1}V_{1} = P_{2}V_{2}$
• Now according to question we assume
  $P_{1} = P$ ,  $V_{1} = V$ then $P_{2} = P'$ and $V_{2} = \frac{V}{4}$
• Now when we put the values in the above equation we get
  $P' = 4P$  ....................(1)
• For polytropically process, we know $PV^Y = Constant$
  $P_{1}V_{1}^Y = P_{2}V_{2}^Y$
  So $P_{1}V_{1}^1^.^5 = P_{2}V_{2}^1^.^5 = Constant$
• Now according to question we assume
  $P_{1} = P$ , $V_{1} = V$ then $P_{2} = P'$ and $V_{2} = \frac{V}{4}$
• By putting these values in the equation we get
  $P' = 8P$  ....................(2)
• By dividing eq. (1) by eq.(2) we get
  $\frac{P_{1}}{P_{2}} = \frac{1}{2}$