RMS and KE Expressions

Q.

The value of Stefans constant is σ = 5.67 × 10-8 J s-m² K^-4. Find its value in CGS system, where joule (J) and erg are the units of energy in SI and CGS system, respectively. [ Hint 1 J = 10⁷ erg ]

Q.

The average molar mass of chlorine is $35.5\mathrm{g}{\mathrm{mol}}^{-1}$. The ratio of ${}^{35}\mathrm{Cl}$ to ${}^{37}\mathrm{Cl}$ in naturally occurring chlorine is close to

1. $1:1$

2. $3:1$

3. $2:1$

4. $4:1$

Q.

If the rms speed of nitrogen molecules is $490m{s}^{-1}$ at 273K, find the rms speed of hydrogen molecules at the same temperature -

1. 1580 m/s

2. 1830 m/s

3. 1700 m/s

4. 2150 m/s

Q. One mole of a diatomic gas undergoes a process $P=\frac{P_o}{1+(\frac{V}{V_o}{)}^3}$ where $P_o$ and $V_o$ are constants. The translational kinetic energy of the gas when $V=V_o$ is given by

1. $\frac{5P_o{V}_o}{4}$
2. $\frac{3P_o{V}_o}{4}$
3. $\frac{3P_o{V}_o}{2}$
4. $\frac{5P_o{V}_o}{2}$

Q.

Molecules of an ideal gas are known to have three translational degrees of freedom and two rotational degrees of freedom. The gas is maintained at a temperature of$T$. The total internal energy, $U$of a mole of this gas, and the value of $\gamma =\left(\frac{C_p}{C_v}\right)$ being given, respectively by:

1. $U=\left(\frac{5}{2}\right)RT,\gamma =\frac{7}{5}$

2. $U=5RT,\gamma =\frac{6}{5}$

3. $U=5RT,\gamma =\frac{7}{5}$

4. $U=\left(\frac{5}{2}\right)RT,\gamma =\frac{6}{5}$

Q.

The average translational kinetic energy of air molecules is 0.040 eV (1 eV = 1.6 $\times {10}^{-19}$ J). Calculate the temperature of the air. Boltzmann constant k = 1.38 $\times 10-23J{K}^{-1}$

Q. At what temperature, the mean kinetic energy of $O_2$ will be the same as $H_2$ molecules at – ${73}^{\circ }$C

1. 127°C
2. 527°C
3. – 73°C
4. – 173°C

Q. The average rotational kinetic energy for the diatomic gas molecules is $(k_B$ is Boltzmann constant and $T$ is absolute temperature)

1. $\frac{3}{2}{k}_{B}T$
2. $\frac{7}{2}{k}_{B}T$
3. $k_{B}T$
4. $\frac{5}{2}{k}_{B}T$

Q. A $2100\text{W}$ continuous flow geyser (instant geyser) has water inlet temperature $={10}^{\circ }C$ while the water flows out at the rate of $20\text{g/sec}$. The outlet temperature of water must be about
(Assume that time lag between inlet and outlet flow is $1\text{s}$)

1. ${20}^{\circ }C$
2. ${30}^{\circ }C$
3. ${35}^{\circ }C$
4. ${40}^{\circ }C$

Q.

Calculate the rms speed of nitrogen at STP (pressure = 1 atm and temperature = $0^{\circ }C$). The density of nitrogen in these conditions is $1.25kg{m}^{-3}$.

1. 490 m/s

2. 510 m/s

3. 410 m/s

4. 370 m/s

Q. How many times a diatomic gas should be expanded adiabatically so as to reduce the root mean square velocity to half?

1. $64$
2. $32$
3. $16$
4. $8$

Q. At which of the following temperature would the molecules of a gas have twice the average kinetic energy they have at ${20}^{\circ }C$

1. 
2. 
3. 
4. 

Q.

When the temperature of a substance increases, what happens to the speed of the molecules?

Q. A vessel is partitioned in two equal halves by a fixed diathermic seperator. Two different ideal gases are filled in the left($L$) and right($R$) halves. The rms speed of the molecules in $L$ part is equal to the mean speed in the $R$ part. Then the ratio of the mass of a molecule in $L$ part to that of a molecule in $R$ part is

1. $\frac{\sqrt{3}}{2}$
2. $\frac{\sqrt{2}}{3}$
3. $\frac{\sqrt{\pi }}{2}$
4. $\frac{3\pi }{8}$

Q.

Kinetic energy of molecules in helium is __________[Large/very large] compared to the kinetic energy of molecules in water.

Q.

What will be the ratio between the root mean square speed of ${\mathrm{H}}_2$ at $50\mathrm{K}$ and that of ${\mathrm{O}}_2$ at $800\mathrm{K}$?

1. $1$

2. $2$

3. $4$

4. $1/4$

Q. A gas mixture consists of $2$ moles of $O_2$ and $4$ moles of Ar at temperature $T$. Neglecting all vibrational modes, the total internal energy of the system is

1. 15 RT
2. 11 RT
3. 9 RT
4. 15 RT

Q.

A gas occupies a volume of 400 cm³. On heating to 127°C its volume becomes 1600 cm³.Find the initial temperature of a gas on Celsius scale. Assume pressure remains constant.

Q.

The Kinetic energy of N molecules of ${\mathrm{H}}_2\mathrm{is}3\mathrm{J}\mathrm{at}-{73}^0\mathrm{C}$ the Kinetic energy of the same sample of ${\mathrm{H}}_2\mathrm{at}-{127}^0\mathrm{C}$ is ?

Q. $O_2$ is $16$ times heavier than $H_2$. If at the same temperature the $O_2$ molecules have average kinetic energy $E,$ then at the same temperature the average kinetic energy of $H_2$ molecules will be

1. $\frac{E}{4}$
2. $4E$
3. $E$
4. $\frac{E}{16}$

Q. A cylindrical container is divided in two equal parts by a diathermic piston. Different ideal gases are filled in the two parts. Find the ratio of the mass of the molecules of the gas in the lower part to that of the upper part, if the root mean square velocity of molecules in the lower part is equal to the mean velocity of molecules in the upper part.

1. $1.224$
2. $1.178$
3. $1.288$
4. $1.128$

Q.

A vessel containing one mole of a monatomic ideal gas (molecular weight = $20gmo{l}^{-1}$ is moving on a floor at aspeed of 50 $m.s^{-1}.$ The vessel is stopped suddenly. Assuiming that the mechnical energy lost has gone into the internal energy of the gas, find the rise in its temperature.

Q.

If $K$ is the Boltzmann constant, the average kinetic energy of a gas molecule at absolute Temperature $T$ is

1. KT

2. KT/2

3. 3KT/2

4. 3K T/4

Q. A gas mixture consists of molecules of type 1, 2 and 3 with molar masses $m_1>m_2>m_3{\nu }_{rms}.\overline{K}$ and are the rms speed and average kinetic energy of the gases. Which of the following is true

1. and
   
2. and
   
   
3. and
   
   
4. and
   
   

Q.

What is the translational kinetic energy of a Helium molecule (He) at 1500 K?

1. 2.189 eV

2. 0.194 eV

3. 1.215 eV

4. 10.068 eV

Q.

At what temperature does the rms speed of oxygen molecules equal the escape velocity of - (i) Earth, (ii) the Moon? Assume gravitational acceleration at the moon is 0.6g.

1. 

2. 

3. 

4. 

Q. One mole of helium is kept in a cylinder of cross-section $A=8.5{\text{cm}}^2$. The cylinder is closed by a light frictionless piston. The gas is slowly heated in a process during which a total of $42\text{J}$ heat is given to the gas. If the temperature rises through $2^{\circ }C$, find the displacement of the piston.
Atmospheric pressure $=100\text{kPa.}$
The internal energy of $n$ moles of a monoatomic ideal gas is $1.5\text{nRT}$.

1. $20\text{cm}$
2. $30\text{cm}$
3. $40\text{cm}$
4. $50\text{cm}$

Q. If the rms velocity of a gas is $v$, then

1. $v^{2}T=\text{constant}$
2. $v^2/T=\text{constant}$
3. $v{T}^2=\text{constant}$
4. $v$ is independent of $T$

Q. 1joule of energy is to be converted into new system of units in which length is measured in 10 metre , mass in 10 kg and time in 1 minute . The numerical value of 1 joule in the new system is ?

Q. Match $\text{Column-I}$ and $\text{Column-II}$ and choose the correct match from the given choices.

$\begin{array}{cc}\text{Column-I}& \text{Column-II}\\ \text{(A) Root mean square}& \text{(P)}\frac{1}{3}nm{\overline{v}}^2\\ \text{speed of gas}& \\ \text{molecules}& \\ \text{(B) Pressure exerted}& \text{(Q)}\sqrt{\frac{3RT}{M}}\\ \text{by ideal gas}& \\ \text{(C) Average kinetic}& \text{(R)}\frac{5}{2}RT\\ \text{Energy of a}& \\ \text{molecule}& \\ \text{(D) Total internal}& \text{(R)}\frac{3}{2}{k}_{B}T\\ \text{energy of 1 mole}& \\ \text{of a diatomic gas}& \end{array}$

1. $\text{(A)-(R), (B)-(Q), (C)-(P), (D)-(S)}$
2. $\text{(A)-(R), (B)-(P), (C)-(S), (D)-(Q)}$
3. $\text{(A)-(Q), (B)-(R), (C)-(S), (D)-(P)}$
4. $\text{(A)-(Q), (B)-(P), (C)-(S), (D)-(R)}$