Question

An ideal gas (γ = 1.5) is expanded adiabatically. How many times has the gas had to be expanded to reduce the root mean square velocity of molecules two times?

Options

• 8 times

• 20 times

• 16 times

• 12 times

Answer 1

16 times

Explanation:

Given, γ = 1.5

rms velocity of the gas molecule,

`"v"_"rms" = sqrt((3RT)/m)` ⇒ T ∝ v²

⇒ `T_2/T_1 = "v"_2^2/"v"_1^2`

Here, v₂ = `"v"_1/2`

∴ `T_2/T_1 = ("v"_1/2)^2/"v"_1^2 = 1/4` ...........(i)

For adiabatic process,

TV^{γ - 1} = constant

⇒ `("V"_2/"V"_1)^{gamma - 1} = (T_1/T_2)`

⇒ `"V"_2/"V"_1 = (T_1/T_2)^{1/(gamma - 1)} = (4)^{1/(1.5 - 1)}` [using Eq. (i)]

= (4)² = 16

Hence, gas has to be expanded to 16 times.