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. \(\frac{1}{4} \frac{P_o V_o}{R}\)