The temperature of an open room of volume \(30~\text{m}^3\) increases from \(17^\circ \text{C}\) to \(27^\circ \text{C}\) due to the sunshine. The atmospheric pressure in the room remains \(1\times 10^{5}~\text{Pa}\). In \(n_i\) and \(n_f\) are the number of molecules in the room before and after heating, the \(n_f\text-n_i \) will be:
1. \( -1.61 \times 10^{23} \)
2. \( 1.38 \times 10^{23} \)
3. \( 2.5 \times 10^{25} \)
4. \( -2.5 \times 10^{25}\)

If \(10^{22}\) gas molecules each of mass \(10^{-22}\) kg collide with a surface (perpendicular to it) elastically per second over an area \(1~\text{m}^2\) with a speed \(10^4~\text{m/s}\), the pressure exerted by the gas molecules will be of the order of:
1. \( 10^8 ~\text{N/m}^2\)
2. \(10^4 ~\text{N/m}^2\)
3. \(10^{16} ~\text{N/m}^2\)
4. \(10^3 ~\text{N/m}^2\)

For a given at \(1\) atm pressure, the rms speed of the molecules is \(200~\text{m/s}\) at \(127^\circ\text{C}.\) At \(2\) atm pressure and at \(227^\circ\text{C},\) the rms speed of the molecules will be:

An HCl molecule has rotational, translational and vibrational motions. If the rms velocity of HCl molecules in its gaseous phase is \(\vec{v}\), \(m\) is its mass and \(k_B\) is Boltzmann constant, then its temperature will be:
1. \( \frac{m v^{2}}{7 k_B} \)
2. \(\frac{m v^2}{6 k_B} \)
3. \(\frac{m {v}^2}{5 k_B} \)
4. \(\frac{m v^2}{3 k_B} \)
 

One mole of an ideal gas undergoes a process in which pressure and volume are related by the equation:
        \(P=P_0\left[1-\dfrac{1}{2}\left(\dfrac{V_0}{V}\right)^2\right] \)
where \(P_0\)​ and \(V_0\)​ are constants. If the volume of the gas increases from \(V=V_0\)​ to \(V=2V_0,\) what is the resulting change in temperature?
1. \( \frac{3}{4} \frac{P_o V_o}{R} \)
2. \(\frac{1}{2} \frac{P_o V_o}{R} \)
3. \(\frac{5}{4} \frac{P_o V_o}{R} \)
4. \(\frac{1}{4} \frac{P_o V_o}{R}\)

A gas mixture consists of \(3\) moles of oxygen and \(5\) moles of argon at temperature \(T.\) Assuming the gases to be ideal and the oxygen bond to be rigid, the total internal energy (in units of \(RT\)) of the mixture is:
1. \(11\)
2. \(15\)
3. \(20\)
4. \(13\)