A gas mixture consists of molecules of type 1, 2 and 3, with molar masses ${\mathrm{m}}_1>{\mathrm{m}}_2>{\mathrm{m}}_3. {\mathrm{V}}_{\mathrm{rms}}$ and $\overline{\mathrm{K}}$ are the r.m.s. speed and average kinetic energy of the gases. Which of the following is true

1. ${\left({\mathrm{V}}_{\mathrm{rms}}\right)}_1<{\left({\mathrm{V}}_{\mathrm{rms}}\right)}_2<{\left({\mathrm{V}}_{\mathrm{rms}}\right)}_3 \mathrm{and} {\left(\overline{\mathrm{K}}\right)}_1={\left(\overline{\mathrm{K}}\right)}_2=\left({\overline{\mathrm{K}}}_3\right)$

2. ${\left({\mathrm{V}}_{\mathrm{rms}}\right)}_1={\left({\mathrm{V}}_{\mathrm{rms}}\right)}_2={\left({\mathrm{V}}_{\mathrm{rms}}\right)}_3 \mathrm{and} {\left(\overline{\mathrm{K}}\right)}_1={\left(\overline{\mathrm{K}}\right)}_2>{\left(\overline{\mathrm{K}}\right)}_3$

3. ${\left({\mathrm{V}}_{\mathrm{rms}}\right)}_1>{\left({\mathrm{V}}_{\mathrm{rms}}\right)}_2>{\left({\mathrm{V}}_{\mathrm{rms}}\right)}_3 \mathrm{and} {\left(\overline{\mathrm{K}}\right)}_1<{\left(\overline{\mathrm{K}}\right)}_2>\left({\overline{\mathrm{K}}}_3\right)$

4. ${\left({\mathrm{V}}_{\mathrm{rms}}\right)}_1>{\left({\mathrm{V}}_{\mathrm{rms}}\right)}_2>{\left({\mathrm{V}}_{\mathrm{rms}}\right)}_3 \mathrm{and} {\left(\overline{\mathrm{K}}\right)}_1<{\left(\overline{\mathrm{K}}\right)}_2<{\left(\overline{\mathrm{K}}\right)}_3$

Two ideal gases at absolute temperature ${\mathrm{T}}_1$ and ${\mathrm{T}}_2$ are mixed. There is no loss of energy. The masses of the molecules are ${\mathrm{m}}_1$ and ${\mathrm{m}}_2$ and the number of molecules in the gases are ${\mathrm{n}}_1$ and ${\mathrm{n}}_2$ respectively. The temperature of mixture will be

1. $\frac{{\mathrm{T}}_1+{\mathrm{T}}_2}{2} \mathrm{energy}$                     

2. $\frac{{\mathrm{T}}_1+{\mathrm{T}}_2}{{\mathrm{n}}_1{\mathrm{n}}_2}$

3. $\frac{{\mathrm{n}}_1{\mathrm{T}}_1+{\mathrm{n}}_2{\mathrm{T}}_2}{{\mathrm{n}}_1+{\mathrm{n}}_2}$                         

4. $\left({\mathrm{T}}_1+{\mathrm{T}}_2\right)$

The molecules of an ideal gas at a certain temperature have

1. Only potential energy

2. Only kinetic energy

3. Potential and kinetic energy both

4. None of the above

The temperature at which the average translational kinetic energy of a molecule is equal to the energy gained by an electron in accelerating from rest through a potential difference of 1 volt is

1.  $4.6\times {10}^3 \mathrm{K}$                 

2.  $11.6\times {10}^3 \mathrm{K}$

3.  $23.2\times {10}^3 \mathrm{K}$               

4.  $7.7\times {10}^3 \mathrm{K}$

The kinetic energy of one mole gas at 300 K temperature, is E. At 400 K temperature kinetic energy is E'. The value of E'/E is

1.  $1.33$                         

2.  $\left(\sqrt{\frac{4}{3}}\right)$

3.  $\frac{16}{9}$                           

4.  $2$

\(N\) molecules each of mass \(m\) of gas \(A\) and \(2N\) molecules each of mass \(2m\) of gas \(B\) are contained in the same vessel at temperature \(T\). The mean square of the velocity of molecules of gas \(B\) is \(v^2\) and the mean square of \(x\) component of the velocity of molecules of gas \(A\) is \(w^2\). The ratio is \(\dfrac{w^2}{v^2}\) is:
1. \(1\)
2. \(2\)
3. \(\dfrac{1}{3}\)
4. \(\dfrac{2}{3}\)

A gas is filled in a cylinder, its temperature is increased by 20% on Kelvin scale and volume is reduced by 10%. How much percentage of the gas will leak out: (Assume pressure is constant)

1. 30%                           

2. 40%

3. 15%                           

4. 25%

The air density at Mount Everest is less than that at the sea level. It is found by mountaineers that for one trip lasting a few hours, the extra oxygen needed by them corresponds to 30,000 cc at sea level (pressure 1 atmosphere, temperature 27°C). Assuming that the temperature around Mount Everest is –73°C and that the oxygen cylinder has capacity of 5.2 litre, the pressure at which ${\mathrm{O}}_2$ be filled (at site) in cylinder is

1.  3.86 atm                     

2.  5.00 atm

3.  5.77 atm                     

4.  1 atm

$\frac{1}{2}$ mole of helium gas is contained in a container at S.T.P. The heat energy needed to double the pressure of the gas, keeping the volume constant (specific heat of the gas $=3 \mathrm{J} {\mathrm{gm}}^{-1} {\mathrm{K}}^{-1}$) is 

1.  3276 J                   

2.  1638 J

3.  819 J                     

4.  409.5 J

The equation of state of a gas is given by $\left(\mathrm{P}+\frac{{\mathrm{aT}}^2}{\mathrm{V}}\right){\mathrm{V}}^{\mathrm{c}}=\left(\mathrm{RT}+\mathrm{b}\right),$ where a, b, c and R are constants. The isotherms can be represented by $\mathrm{P}={\mathrm{AV}}^{\mathrm{m}}-{\mathrm{BV}}^{\mathrm{n}},$ where A and B depend only on temperature and 

1. $\mathrm{m}=-\mathrm{c} \mathrm{and} \mathrm{n}=-1$                       

2. $\mathrm{m}=\mathrm{c} \mathrm{and} \mathrm{n}=1$

3. $\mathrm{m}=-\mathrm{c} \mathrm{and} \mathrm{n}=1$                           

4. $\mathrm{m}=\mathrm{c} \mathrm{and} \mathrm{n}=-1$