Solveeit Logo
Login

Question

Physics Question on Adiabatic Processes

The volume of an ideal gas (γ=1.5\gamma = 1.5) is changed adiabatically from 5 litres to 4 litres. The ratio of initial pressure to final pressure is:

A

45\frac{4}{5}

B

1625\frac{16}{25}

C

855\frac{8}{5\sqrt{5}}

D

25\frac{2}{\sqrt{5}}

Answer

855\frac{8}{5\sqrt{5}}

Explanation

Solution

For an adiabatic process, the relation between pressure and volume is:

PiViγ=PfVfγ,P_iV_i^\gamma = P_fV_f^\gamma, where γ=1.5\gamma = 1.5.

Substitute the given volumes (Vi=5 litres,Vf=4 litresV_i = 5 \, \text{litres}, V_f = 4 \, \text{litres}):

Pi(5)1.5=Pf(4)1.5.P_i (5)^{1.5} = P_f (4)^{1.5}.

Rearranging for PiPf\frac{P_i}{P_f}:

PiPf=(4)1.5(5)1.5.\frac{P_i}{P_f} = \frac{(4)^{1.5}}{(5)^{1.5}}.

Simplify:

PiPf=(45)1.5.\frac{P_i}{P_f} = \left(\frac{4}{5}\right)^{1.5}.

Write 1.51.5 as 32\frac{3}{2}:

PiPf=(45)32=(45)1×(45)12.\frac{P_i}{P_f} = \left(\frac{4}{5}\right)^{\frac{3}{2}} = \left(\frac{4}{5}\right)^1 \times \left(\frac{4}{5}\right)^{\frac{1}{2}}.

Simplify each term:

PiPf=45×45=45×25=855.\frac{P_i}{P_f} = \frac{4}{5} \times \sqrt{\frac{4}{5}} = \frac{4}{5} \times \frac{2}{\sqrt{5}} = \frac{8}{5\sqrt{5}}.
Final Answer: 855\frac{8}{5\sqrt{5}}.