Which of the following relations is correct for the root-mean-square speed of gas at temperature T?

Where, k= Boltzmann constant, and m= mass of a molecule.

(1) $V_{r.m.s}=\sqrt{\frac{3<mspace\; width="1em"></mspace>k\hspace{0.17em}T}{m}}$   

(2)$V_{r.m.s}=\sqrt{\frac{8<mspace\; width="1em"></mspace>k\hspace{0.17em}T}{m<mspace\; width="1em"></mspace>\mathrm{\pi }}}$

(3) $V_{r.m.s}=\sqrt{\frac{2<mspace\; width="1em"></mspace>k\hspace{0.17em}T}{m}}$   

(4) $V_{r.m.s}=\sqrt{\frac{m}{3<mspace\; width="1em"></mspace>k<mspace\; width="1em"></mspace>T}}$

 

The average kinetic energy of a helium atom at $30°C$ is:                                                                                           [MP PMT 2004]

1. Less than 1 eV

2. A few KeV

3. 50-60 eV

4. 13.6 eV

The mean translational kinetic energy of a perfect gas molecule at the temperature T Kelvin is:

1. $\frac{1}{2}kT$         

2. kT             

3. $\frac{3}{2}kT$           

4. 2 kT

The curve between absolute temperature and \({v}^2_{rms}\)${\mathrm{}}^{}$ is:

1.		2.
3.		4.

Five molecules of a gas have speeds, 1, 2, 3, 4 and 5 km$s^{-1}$.  The value of the root mean square speed of the gas molecules is:

1. 3 km$s^{-1}$           

2. $\sqrt{11}$ km$s^{-1}$

3. $\frac{1}{2}\sqrt{\mathrm{\pi }}$ km$s^{-1}$   

4. 3.5 km$s^{-1}$

The average velocity of an ideal gas molecule is:

At constant temperature, on increasing the pressure of a gas by 5% , its volume will decrease by:          

1. 5%       

2. 5.26 %         

3. 4.26 %         

4. 4.76 %

The average kinetic energy of a gas molecule can be determined by knowing:                                        [RPET 2000, MP PET 2010]

1. The number of molecules in the gas

2. The pressure of the gas only

3. The temperature of the gas only

4. None  of the above is enough by itself

Volume, pressure, and temperature of an ideal gas are \(V,\) \(P,\) and \(T\) respectively. If the mass of its molecule is \(m,\) then its density is:
[\(k\)=Boltzmann's constant]

One liter of gas A and two liters of gas B, both having the same temperature 100$°$C and the same pressure 2.5 bar will have the ratio of average kinetic energies of their molecules as:

1. 1:1       

2. 1:2     

3. 1:4       

4. 4:1