You visited us 1 times! Enjoying our articles? Unlock Full Access!

Byju's Answer

Standard XII

Chemistry

Root Mean Square Velocity

For gaseous s...

Question

For gaseous state, if the most probable speed is denoted by $C^{*}$ , average speed by $¯ C$ and root mean square speed by $C$ , then for a large number of molecules, the ratios of these speeds are:

\begin{matrix} C^{*} & ¯ C & C 1.225 & 1.128 & 1 \end{matrix}

No worries! We‘ve got your back. Try BYJU‘S free classes today!

\begin{matrix} C^{*} & ¯ C & C 1 & 1.128 & 1.225 \end{matrix}

Right on! Give the BNAT exam to get a 100% scholarship for BYJUS courses

\begin{matrix} C^{*} & ¯ C & C 1.128 & 1.225 & 1 \end{matrix}

No worries! We‘ve got your back. Try BYJU‘S free classes today!

\begin{matrix} C^{*} & ¯ C & C 1 & 1.125 & 1.128 \end{matrix}

No worries! We‘ve got your back. Try BYJU‘S free classes today!

Open in App

Solution

The correct option is B $\begin{matrix} C^{} & ¯ C & C 1 & 1.128 & 1.225 \end{matrix}$
Using thr formula,
$C^{} = \sqrt{\frac{2 R T}{M}}; ¯ C = \sqrt{\frac{8 R T}{π M}}; C = \sqrt{\frac{3 R T}{M}}$
$C^{*} : ¯ C : C = \sqrt{2} : \sqrt{\frac{8}{π}} : \sqrt{3} = 1.414 : 1.596 : 1.732$
$= 1 : 1.128 : 1.225$

Theory:

Kinetic gas equation :

$P V = \frac{1}{3} m N {¯ u}^{2}$

$V o l u m e o f c o n t a i n e r (V) = l^{3}$

Verification of gas laws with kinetic gas equation :

$P V = \frac{1}{3} m N {¯ u}^{2}$ .......................(1)

According to Kinetic theory of gases :

$\frac{1}{2} m {¯ u}_{r m s}^{2} = λ T (i n k e l v i n)$ .................(2)

where $λ$ is a proportionality constant.

Multiply and divide equation 1 by 2 on both sides :

$P V = \frac{2}{3} \frac{1}{2} m N {¯ u}^{2}$ ................(3)

From equation (2) $\frac{1}{2} m {¯ u}_{r m s}^{2} = λ T$ .......(4)

Putting the value of equation 4 in equation 3 :

$P V = \frac{2}{3} (λ T) N$

$P \propto \frac{1}{V}$ which is Boyle's law at constant N and T.

Similarly we can prove charles’ law from kinetic gas equation :

At constant N and Pressure(P):

$V \propto T$ and similarly it can be done at constant N and Volume (V) which is Gay lussac’s law :

$P \propto T$

At constant temperature (T) and Pressure (P) Kinetic gas equation can prove Avogadro’s law :

$V \propto N$ or $V = k_{3} N$

Dividing and multiplying by $N_{A}$

$V = \frac{k_{3}}{N_{A}} (\frac{N}{N_{A}})$ or $V = k_{4} (\frac{N}{N_{A}})$

Calculation of $λ$ :

From ideal gases :

$P V = n R T = \frac{2}{3} λ N T$ ........(1)

Where $n = n u m b e r o f m o l e s o f g a s a n d N_{A} = A v o g a d r o^{'} s n u m b e r$

Since $n u m b e r o f m o l e s (n) = \frac{G i v e n n u m b e r o f m o l e c u l e s (N)}{A v o g a d r o^{'} s n u m b e r (N_{A})}$

$N = n N_{A}$ ...........(2)

Putting the value of N from 2 in 1 :

$P V = n R T = \frac{2}{3} (n N_{A}) λ T$

$λ = \frac{3}{2} \frac{R}{N_{A}}$ or $λ = \frac{3}{2} k$ where k is Boltzmann constant

$k (B o l t z m a n n c o n s t a n t) = \frac{R}{N_{A}}$

Average kinetic energy of molecules :

$\frac{1}{2} m {¯ u}_{r m s}^{2} \propto T (i n k e l v i n)$ or $\frac{1}{2} m {¯ u}_{r m s}^{2} = λ T (i n k e l v i n)$ where $λ$ is a proportionality constant.

$λ T = \frac{3}{2} k T$ where k is Boltzmann constant

$k = 1.38 \times 10^{- 23} J K^{-} 1$

Total Kinetic energy per mole :

$T o t a l k i n e t i c e n e r g y = N_{A} (\frac{1}{2} m {¯ u}_{r m s}^{2}) = N_{A} (λ T)$

$= \frac{3}{2} (\frac{R}{N_{A}}) N_{A} = \frac{3}{2} R$

Root mean square speed $V_{r m s}$ :

$T o t a l k i n e t i c e n e r g y = \frac{1}{2} M {¯ u}_{r m s}^{2} = \frac{3}{2} R T$

${¯ u}_{r m s}^{2} = \frac{3 R T}{M}$ or ${¯ u}_{r m s} = \sqrt[2]{\frac{3 R T}{M}}$

Unit of $u_{r m s}$ :

$Unit of u_{r m s} = \sqrt{\frac{(J K^{- 1} m o l^{- 1}) (K)}{k g m o l^{- 1}}}$

$Unit of u_{r m s} = \sqrt{\frac{J}{k g}}$

$Unit of u_{r m s} = \sqrt{\frac{k g m^{2} s^{- 2}}{k g}}$

$Unit of u_{r m s} = m s^{- 1}$

Note: Always take $R$ in $J K^{- 1} m o l^{- 1}$ and $M$ in $k g m o l^{- 1}$

Suggest Corrections

Similar questions

Q. For Gaseous state, if most probable speed is denoted by

C^{+}

, average speed by

¯ ¯¯ ¯ C

and mean square speed by

C

$, then for a large number of molecules the ratios of these speeds are:

For gaseous state, if most probale speed is denoted by C* , average speed by and mean square speed by $¯ C$ , then for a large number of molecules the ratios of these speeds are

Q. For gaseous state, if most probable speed is denoted by average speed by

¯ ¯¯ ¯ C

and mean square speed by

C

, then for large number of molecules the ratios of these speeds are:

Q. For a gaseous state, if most probable speed is denoted by

C^{*}

, average speed by

¯ ¯¯ ¯ C

and mean square by

C

, then for a large number of molecules the ratios of these speeds are:

Q. For gaseous state, if most probable speed is denoated by

C^{*}

, average speed by

¯ ¯¯ ¯ C

and mean square speed by

C

, then for a large number of molecules the ratios of these speeds are:

Join BYJU'S Learning Program

Explore more

Root Mean Square Velocity

Standard XII Chemistry

Join BYJU'S Learning Program

For gaseous state, if the most probable speed is denoted by C∗, average speed by ¯C and root mean square speed by C, then for a large number of molecules, the ratios of these speeds are:

For gaseous state, if the most probable speed is denoted by $C^{*}$ , average speed by $¯ C$ and root mean square speed by $C$ , then for a large number of molecules, the ratios of these speeds are: