Root Mean Square Velocity

Q. The root mean square speed of molecules of nitrogen gas is $v$ at a certain temperature. When the temperature is doubled, the molecules dissociate into individual atoms. The new rms speed of the atom is:

1. $\sqrt{2}v$
2. $2v$
3. $v$
4. $4v$

Q. The molar mass of a certain gas $B$ is double than that of $A$. Also the RMS velocity of gas $A$ is $200{\text{ms}}^{-1}$ at a certain temperature. RMS velocity of gas $B$ at the temperature half of the temperature of $A$ is:

1. $200{\text{ms}}^{-1}$
2. $300{\text{ms}}^{-1}$
3. $400{\text{ms}}^{-1}$
4. $100{\text{ms}}^{-1}$

Q. In a container of capacity 1 litre there are ${10}^{23}$ molecules each of mass ${10}^{-22}\text{g}$. If the root mean square speed is ${10}^5\text{cm/s}$ then, calculate the pressure of the gas.

1. $1\text{atm}$
2. $1\text{torr}$
3. $\frac{1}{3}\times {10}^7\text{Pa}$
4. $1\times {10}^7\text{Pa}$

Q. A vessel of capactity $1d{m}^3$ contains $1.03\times {10}^{23}{H}_2$ molecules exerting a pressure of $101.325\text{kPa}$. Calculate RMS speed and average speed.

1. $\begin{array}{cc}\text{RMS speed}& \text{Average speed}\\ 942.5{\text{m s}}^{-1}& 868.35{\text{m s}}^{-1}\end{array}$
2. $\begin{array}{cc}\text{RMS speed}& \text{Average speed}\\ 942.5{\text{m s}}^{-1}& 722.12{\text{m s}}^{-1}\end{array}$
3. $\begin{array}{cc}\text{RMS speed}& \text{Average speed}\\ 942.5{\text{m s}}^{-1}& 622.53{\text{m s}}^{-1}\end{array}$
4. $\begin{array}{cc}\text{RMS speed}& \text{Average speed}\\ 868.36{\text{m s}}^{-1}& 942.5{\text{m s}}^{-1}\end{array}$

Q. The average velocity of gas molecules is $400\text{m/sec}$. Calculate its rms velocity at the same temperature.

Q. The density of a gas filled in an electric lamp is $0.75\text{kg}/{\text{m}}^3$ . After the lamp has been switched on, the pressure in it increases from $4\times {10}^4\text{Pa}$ to $9\times {10}^4\text{Pa}$. What is the increase in $U_{RMS}$ ?

1. 100 m/s
2. 200 m/s
3. 300 m/s
4. None of these

Q. A flask has 10 gas particles out of which four particles are moving at $7{\text{ms}}^{-1}$ and all the remaining ones are moving at the speed of $\text{'}x\text{'}{\text{ms}}^{-1}$. If the $V_{rms}$ of the gas is $5{\text{ms}}^{-1}$, what is the value of $‘x^{\prime }$ ?

Q. At what temperature will the r.m.s velocity of oxygen be 3/2 times of its value at S.T.P.?

1. 614.25 K
2. 674.25 K
3. 914.25 K
4. 974 K

Q. Temperature at which the rms speed of $O_2$ is equal to that of neon at $300\text{K}$ is:

1. $280\text{K}$
2. $680\text{K}$
3. $180\text{K}$
4. $480\text{K}$

Q. The root mean square velocity of hydrogen is $\sqrt{5}$ times than that of nitrogen. If $T$ is the temperature of the gas, then:

1. $T_{H_2}=T_{N_2}$
2. $T_{H_2}>T_{N_2}$
3. $T_{H_2}<T_{N_2}$
4. $T_{H_2}=\sqrt{7}{T}_{N_2}$

Q. The root mean square speed of $8g$ of $He$ is $300{\text{ms}}^{-1}$. Total kinetic energy of $He$ gas is:

1. $120J$
2. $240J$
3. $360J$
4. None of these

Q. For $C{O}_2$, given that average velocity at $T_1$ is equal to most probable velocity at $T_2$. Thus, $\frac{T_1}{T_2}$

1. 1.27
2. 0.78
3. 1.13
4. 2.26

Q. A gas 1g at 4 bar pressure. If we add 2g of gas B then the total pressure inside the container is 6 bar. Which of the following is true ?

1. $M_A=2M_B$
2. $M_B=2M_A$
3. $M_A=4M_B$
4. $M_B=4M_A$

Q. Temperature at which the rms speed of $O_2$ is equal to that of neon at $300\text{K}$ is:

1. $280\text{K}$
2. $480\text{K}$
3. $680\text{K}$
4. $180\text{K}$

Q. In a container of capacity 1 litre, there are ${10}^{23}$ molecules each of mass ${10}^{-22}\text{g}$. If the root mean square speed is ${10}^5\text{cm/s}$ then calculate the pressure of the gas.

1. $\frac{1}{3}\times {10}^7\text{Pa}$
2. $1\times {10}^7\text{Pa}$
3. $1\text{atm}$
4. $1\text{torr}$

Q. At $298\text{K}$, which of the following gases has the lowest average molecular speed ?

1. $C{H}_4$ at $0.80\text{atm}$
2. $He$ at $0.40\text{atm}$
3. $C{O}_2$ at $0.20\text{atm}$
4. $NO$ at $1.00\text{atm}$

Q. For $C{O}_2$, given that average velocity at $T_1$ is equal to most probable velocity at $T_2$. Thus, $\frac{T_1}{T_2}$

1. 2.26
2. 1.27
3. 0.78
4. 1.13

Q. Two containers are maintained at the same temperature. The first one has a volume of 8 litres and stores hydrogen at 2 atm. The other one has a volume of 4 litres and stores oxygen at 1 atm. What is the ratio of the root mean square velocities of hydrogen and oxygen?

1. 4:1
2. 1:4
3. 1:16
4. 16:1

Q. Distribution of fraction of molecules with velocity is represented in the figure.

Velocity corresponding to point $X$ is:

1. $\frac{\sqrt{2}RT}{M}$
2. $\frac{\sqrt{3}RT}{M}$
3. $\frac{\sqrt{8}RT}{M}$
4. None

Q. A vessel of capactity $1d{m}^3$ contains $1.03\times {10}^{23}{H}_2$ molecules exerting a pressure of $101.325\text{kPa}$. Calculate RMS speed and average speed.

1. $\begin{array}{cc}\text{RMS speed}& \text{Average speed}\\ 942.5{\text{m s}}^{-1}& 868.35{\text{m s}}^{-1}\end{array}$
2. $\begin{array}{cc}\text{RMS speed}& \text{Average speed}\\ 942.5{\text{m s}}^{-1}& 722.12{\text{m s}}^{-1}\end{array}$
3. $\begin{array}{cc}\text{RMS speed}& \text{Average speed}\\ 868.36{\text{m s}}^{-1}& 942.5{\text{m s}}^{-1}\end{array}$
4. $\begin{array}{cc}\text{RMS speed}& \text{Average speed}\\ 942.5{\text{m s}}^{-1}& 622.53{\text{m s}}^{-1}\end{array}$

Q. Identify the correct statement(s).

1. For a gas in a rigid container, mean free path is independent of pressure and temperature
2. Translational kinetic energy of a given amount of an ideal gas is independent of pressure
3. All the molecules of a gas at a given temperature have the same kinetic energy
4. The average kinetic energy of $1\text{mol}$ of gas molecules at a given temperature is proportional to ${}^{\prime }{M}^{\prime }$. (where '$M$' is molar mass of gas)

Q. The kinetic energy of a molecule of a gas is directly proportional to the _______ of the gas.

1. pressure of the gas
2. absolute temperature
3. mass of the gas
4. none of these

Q. At same temperature, calculate the ratio of average velocities of $S{O}_2$ and $C{H}_4$:

1. 2 : 3
2. 3 : 4
3. 1 : 2
4. 1 : 6

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

1. $\begin{array}{ccc}{C}^{\ast }& \overline{C}& C\\ 1.225& 1.128& 1\end{array}$
2. $\begin{array}{ccc}{C}^{\ast }& \overline{C}& C\\ 1& 1.128& 1.225\end{array}$
3. $\begin{array}{ccc}{C}^{\ast }& \overline{C}& C\\ 1.128& 1.225& 1\end{array}$
4. $\begin{array}{ccc}{C}^{\ast }& \overline{C}& C\\ 1& 1.125& 1.128\end{array}$

Q. The molar mass of a certain gas $B$ is double than that of $A$. Also the RMS velocity of gas $A$ is $200{\text{ms}}^{-1}$ at a certain temperature. RMS velocity of gas $B$ at the temperature half of the temperature of $A$ is:

1. $200{\text{ms}}^{-1}$
2. $300{\text{ms}}^{-1}$
3. $400{\text{ms}}^{-1}$
4. $100{\text{ms}}^{-1}$

Q. The ratio of the root mean square speed of $H_2$ gas at $50\text{K}$ and that of $O_2$ gas at $800\text{K}$ is:

1. 4
2. 2
3. 1
4. 0.25

Q. Which of the following gaseous fuels has the lowest calorific value?

1. Refinery gas
2. Gober gas
3. Converter gas
4. Blast furnace gas

Q. In a container of capacity 1 litre, there are ${10}^{23}$ molecules each of mass ${10}^{-22}\text{g}$. If the root mean square speed is ${10}^5\text{cm/s}$ then calculate the pressure of the gas.

1. $\frac{1}{3}\times {10}^7\text{Pa}$
2. $1\times {10}^7\text{Pa}$
3. $1\text{atm}$
4. $1\text{torr}$

Q. For $C{O}_2$, given that average velocity at $T_1$ is equal to most probable velocity at $T_2$. Thus, $\frac{T_1}{T_2}$

1. 2.26
2. 1.27
3. 0.78
4. 1.13

Q. For gaseous state, most probable speed is denoted by $C^{\ast }$, average speed by $\overline{C}$ and root mean square speed by $C$ respectively. For the same number of molecules at same temperature, the ratios of these speeds are:

1. $C^{\ast }:\overline{C}:C=1.225:1.128:1$
2. $C^{\ast }:\overline{C}:C=1.128:1.225:1$
3. $C^{\ast }:\overline{C}:C=1:1.128:1.225$
4. $C^{\ast }:\overline{C}:C=1:1.225:1.128$