An ideal gas (γ = 1 .5) is expanded adiabatically. How many times its initial volume should be expanded to reduce the root-mean-square velocity of molecules by a factor of text2 ?

Question

An ideal gas (&gamma; = 1 .5) is expanded adiabatically. How many times its initial volume should be expanded to reduce the root-mean-square velocity of molecules by a factor of text2 ? (A)  text4 times (B)  text16 times (C)  text8 times (D)  text2 times

Tardigrade Admin · Accepted Answer

vrms=√(3 RT/M)orvrms propto √T vrms is to reduce two times, i.e. the temperature of the gas will have to reduce four times or (T '/T)=(1/4) During the adiabatic process, T Vâ¡&gamma; - 1=Tâ¡'Vâ¡' &gamma; - 1 or (V '/V)=((T/T'))(1/&gamma; - 1) =(4)(1/1.5 - 1)=42=16 ∴ V'=16 V

Q. An ideal gas $(γ = 1.5)$ is expanded adiabatically. How many times its initial volume should be expanded to reduce the root-mean-square velocity of molecules by a factor of $2$ ?

$4$ times

$16$ times

$8$ times

$2$ times

Q. An ideal gas (γ=1.5) is expanded adiabatically. How many times its initial volume should be expanded to reduce the root-mean-square velocity of molecules by a factor of 2 ?

4 times

16 times

8 times

2 times

vrms​=M3RT​​orvrms​∝T​ vrms​ is to reduce two times, i.e. the temperature of the gas will have to reduce four times or TT′​=41​ During the adiabatic process, TV⁡γ−1=T⁡′V⁡′γ−1 or VV′​=(T′T​)γ−11​ =(4)1.5−11​=42=16 ∴ V′=16V

Q. An ideal gas $(γ = 1.5)$ is expanded adiabatically. How many times its initial volume should be expanded to reduce the root-mean-square velocity of molecules by a factor of $2$ ?

$4$ times

$16$ times

$8$ times

$2$ times