Tag: speed of sound in gas

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Multiple choice speed of sound in gas speed of a travelling wave oscillation and waves waves physics

If the pressure of a fixed quantity of a gas is increased 4 times keeping the temperature constant, the r.m.s velocity will :

  1. get doubled

  2. get halved

  3. remain same

  4. get quadrupled
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

RMS velocity is independent of pressure. Hence velocity does not change with variations in pressure

The correct option is (c)

Multiple choice speed of sound in gas speed of a travelling wave oscillation and waves waves physics

If $C _{s}$ be the velocity of sound in air and $C$ be the rms velocity, then

  1. $C _{S} < C$
  2. $C _{s}=c$
  3. $C _{s}=C\left(\dfrac {\gamma}{3}\right)^{1/2}$
  4. $None\ of\ these$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Speed of sound in air, ${{C} _{s}}=\sqrt{\dfrac{\gamma P}{\rho }}\,\ldots \ldots \,(1)$

 $ Where, $

$ \gamma =specific\,heat\,ratio $

$ P=\,pressure $

$ \rho =\,density $

RMS velocity of air molecule, $C=\sqrt{\dfrac{3\overline{R}T}{{{M} _{o}}}}=\sqrt{\dfrac{3P}{\rho }}\,\ldots \ldots \,(2)$

$ where,\, $

$ \overline{R}=\text{universal}\,\text{gas}\,\text{constant} $

$ {{M} _{o}}=Molecular\,mass $

$ T=temperature $

From (1) and (2)

${{C} _{s}}=C{{\left( \dfrac{\gamma }{3} \right)}^{1/2}}$ 

Multiple choice speed of sound in gas speed of a travelling wave oscillation and waves waves physics

With increase in temperature, the rms speed and wave speed in a gas

  1. increases with temperature

  2. decreases with temperature

  3. are independent of temperature

  4. none of the above
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Both RMS speed and speed of sound in gas are directly proportional to temperature. Thus, both the speeds increases with temperature

The correct option is (a)

Multiple choice speed of sound in gas speed of a travelling wave oscillation and waves waves physics

If nitrogen gas molecule goes straight up with its rms speed at $0^o$C from the surface of the earth and there are no collisions with other molecules, then it will rise to an approximate height of:

  1. $18$ km
  2. $15$ km
  3. $12.38$ km
  4. $8$ km
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Molecular mass of Nitrogen molecule=$14$ g/mol

As nitrogen exists as ${N} _{2}$=28 g/mol=$0.028$ kg/mol
Also we know ${ v } _{ rms }=\sqrt { \dfrac { 3RT }{ M }  } $  where R= gas constant=8.31 bar/(K mol)=8.31$\times{10}^{5}$ Pa/(K mol)
T= temperature=${0}^{0}$ C=273 K
Also height $=\dfrac{{V}^{2} _{rms}}{2g}$

$=\dfrac { 3\times 8.31\times { 10 }^{ 5 }\times 273 }{ 2\times 9.81\times 0.028 } \ =12388\quad m=12.38\quad km$

Multiple choice speed of sound in gas speed of a travelling wave oscillation and waves waves physics

If $v _{rms}$ = root mean square speed of molecules
$v _{av}$ = average speed of molecules
$v _{mp}$ = most probable speed of molecules
Then, identify the correct relation between these speeds.

  1. $v _{rms} > v _{av} > v _{mp} $
  2. $v _{av} > v _{mp} > v _{rms}$
  3. $v _{mp} > v _{av} > v _{rms} $
  4. $v _{rms} > v _{av} = v _{mp}$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

root mean square speed of molecules > average speed of molecules > most probable speed of molecules 

so the answer is A.

Multiple choice speed of sound in gas speed of a travelling wave oscillation and waves waves physics

The velocity of sound at the same pressure in two monoatomic gases of densities $ \rho _1$  and $\rho _2$ are $v _1$ and $v _2 $ respectively. If $ \dfrac {\rho _1}{\rho _2} = 4 $ then the value of $ \dfrac {v _1}{v _2} $ is:-

  1. $ \dfrac {1}{4} $
  2. $ \dfrac {1}{2} $
  3. $2$
  4. $4$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Velocity v = sqrt(gamma * P / rho). Since pressure P and gamma are constant, v is inversely proportional to sqrt(rho). Thus, v1 / v2 = sqrt(rho2 / rho1). Given rho1 / rho2 = 4, then rho2 / rho1 = 1/4. So v1 / v2 = sqrt(1/4) = 1/2.

Multiple choice speed of sound in gas speed of a travelling wave oscillation and waves waves physics

Two moles of hydrogen are mixed with n moles of helium. The root mean square speed of gas molecules in the mixture is $\sqrt2$ times the speed of sound in the mixture. Then n is 

  1. $3$
  2. $2$
  3. $1.5$
  4. $2.5$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

v_rms = sqrt(3RT/M_mix). v_sound = sqrt(gamma_mix * RT / M_mix). Given v_rms = sqrt(2) * v_sound, then 3RT/M_mix = 2 * gamma_mix * RT / M_mix, so gamma_mix = 1.5. For a mixture, gamma = (n1Cp1 + n2Cp2) / (n1Cv1 + n2Cv2). With 2 moles H2 (gamma=1.4, Cv=2.5R) and n moles He (gamma=1.67, Cv=1.5R), solving for n yields 2.

Multiple choice speed of sound in gas speed of a travelling wave oscillation and waves waves physics

Two moles of helium are mixed with $n$ moles of hydrogen. The root mean square $\left( rms \right) $ speed of gas molecules in the mixture is $\sqrt { 2 } $ times the speed of sound in the mixture. Then, the value of $n$ is

  1. $1$
  2. $3$
  3. $2$
  4. ${ 3 }/{ 2 }$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

$\because { v } _{ rms }=\sqrt { \dfrac { 3RT }{ M }  } $ and ${ v } _{ sound }=\sqrt { \dfrac { \gamma RT }{ M }  } $,
${ v } _{ rms }=2{ v } _{ sound }$
i.e. $\gamma =\dfrac { 3 }{ 2 } =$ ratio of $\dfrac { { C } _{ p } }{ { C } _{ V } } $ for the mixture
${ C } _{ V }=\dfrac { { n } _{ 1 }{ C } _{ { V } _{ 1 } }+{ n } _{ 2 }{ C } _{ { V } _{ 2 } } }{ { n } _{ 1 }+{ n } _{ 2 } } $
and ${ C } _{ p }=\dfrac { { n } _{ 1 }{ c } _{ { p } _{ 1 } }+{ n } _{ 2 }{ C } _{ { p } _{ 2 } } }{ { n } _{ 1 }+{ n } _{ 2 } } $
$\therefore \gamma =\dfrac { { C } _{ p } }{ { C } _{ V } } =\dfrac { { n } _{ 1 }{ C } _{ { p } _{ 1 } }+{ n } _{ 2 }{ C } _{ { p } _{ 2 } } }{ { n } _{ 1 }{ C } _{ { V } _{ 1 } }+{ n } _{ 2 }{ C } _{ { V } _{ 2 } } } $
$\therefore \dfrac { 3 }{ 2 } =\dfrac { 2\left( \dfrac { 5 }{ 2 } R \right) +n\left( \dfrac { 7 }{ 2 } R \right)  }{ 2\left( \dfrac { 3 }{ 2 } R \right) +n\left( \dfrac { 5 }{ 2 } R \right)  } $
$\Rightarrow \dfrac { 3 }{ 2 } =\dfrac { 10+7n }{ 6+5n } $
$\Rightarrow n=2$