The ratio of ${\mathrm{v}}_{\mathrm{rms}}:{\mathrm{v}}_{\mathrm{mp}}:{\mathrm{v}}_{\mathrm{avg}}$ is: (symbols have their usual meaning)

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

The volume and temperature graph is given in the figure below. If pressures for the two processes are different, then which one,  of the following, is true?

The root mean square velocity of the molecules of a gas is 300 m/s. What will be the root mean square speed of the molecules if the atomic weight is doubled and absolute temperature is halved?

Two chambers of different volumes, one containing ${\mathrm{m}}_1$g of a gas at pressure ${\mathrm{P}}_1$ and other containing ${\mathrm{m}}_2$g of the same gas at pressure ${\mathrm{P}}_2$ are joined to each other. If the temperature of the gas remains constant, the common pressure reached is:

1.  $\frac{{\mathrm{m}}_1{\mathrm{P}}_1+{\mathrm{m}}_2{\mathrm{P}}_2}{{\mathrm{m}}_1 + {\mathrm{m}}_2}$

2.  $\frac{{\mathrm{m}}_1{\mathrm{P}}_2+{\mathrm{m}}_2{\mathrm{P}}_1}{{\mathrm{m}}_1 + {\mathrm{m}}_2}$

3.  $\frac{{\mathrm{m}}_1{\mathrm{P}}_2+\left({\mathrm{P}}_1+{\mathrm{P}}_2\right)}{{{\mathrm{m}}^2}_1 + {\mathrm{m}}_2}$

4.  $\frac{\left({\mathrm{m}}_1+{\mathrm{m}}_2\right){\mathrm{P}}_1{\mathrm{P}}_2}{{\mathrm{m}}_1{\mathrm{P}}_2 + {\mathrm{m}}_2{\mathrm{P}}_1}$

At a pressure of 24 × 10⁵ dyne/cm², the volume of O₂ is 10 litre and mass is 20g. The rms velocity will be:

The volume \(V\) versus temperature \(T\) graph for a certain amount of a perfect gas at two pressures \(\mathrm{P}_1\) and $$
\(\mathrm{P}_2\) are shown in the figure. Here:

We have two vessels of equal volume, one filled with hydrogen and the other with equal mass of helium. The common temperature is \(27^{\circ}\mathrm{C}\) . What is the relative number of molecules in the two vessels?
1. \(\frac{n_H}{n_{He}} = \frac{1}{1}\)
2. \(\frac{n_H}{n_{He}} = \frac{5}{1}\)
3. \(\frac{n_H}{n_{He}} = \frac{2}{1}\)
4. \(\frac{n_H}{n_{He}} = \frac{3}{1}\)

During an experiment, an ideal gas is found to obey an additional law VP² = constant. The gas is initially at temperature T and volume V. What will be the temperature of the gas when it expands to a volume 2V?

1. $\mathrm{T}\text{'}=\sqrt{4}\mathrm{T}$

2. $\mathrm{T}\text{'}=\sqrt{2}\mathrm{T}$

3. $\mathrm{T}\text{'}=\sqrt{5}\mathrm{T}$

4. $\mathrm{T}\text{'}=\sqrt{6}\mathrm{T}$

The rms speed of oxygen atoms is v. If the temperature is halved and the oxygen atoms combine to form oxygen molecules, then the rms speed will be:

1. $\frac{v}{\sqrt{2}}$

2. $v\sqrt{2}$

3. 2v

4. $\frac{v}{2}$

Two thermally insulated vessels \(1\) and \(2\) are filled with air at temperatures \(\mathrm{T_1},\) \(\mathrm{T_2},\) volume \(\mathrm{V_1},\) \(\mathrm{V_2}\) and pressure \(\mathrm{P_1},\) \(\mathrm{P_2}\) respectively. If the valve joining the two vessels is opened, the temperature inside the vessel at equilibrium will be:
1. \(T_1+T_2\)

2. \(\frac{T_1+T_2}{2}\)

3. \(\frac{T_1T_2(P_1V_1+P_2V_2)}{P_1V_1T_2+P_2V_2T_1}\)

4. \(\frac{T_1T_2(P_1V_1+P_2V_2)}{P_1V_1T_1+P_2V_2T_2}\)