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:

1.	\(100~\text{m/s}\)	2.	\(80\sqrt{5}~\text{m/s}\)
3.	\(100\sqrt{5}~\text{m/s}\)	4.	\(80~\text{m/s}\)

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 \(25\times 10^{-3}~\text{m}^3\) volume cylinder is filled with \(1\) mol of \(\mathrm{O_2}\) gas at room temperature \((300~\text{K})\). The molecular diameter of \(\mathrm{O_2}\), and its root mean square speed, are found to be \(0.3~\text{nm}\) and \(200~\text{m/s}\), respectively. What is the average collision rate (per second) for an \(\mathrm{O_2}\) molecule?
1. \( \sim 10^{12} \)
2. \( \sim 10^{10} \)
3. \( \sim 10^{13} \)
4. None of these

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. \(13\)

An ideal gas is confined in a closed container and slowly heated. As the temperature rises, which of the following statements are correct?

Consider a gas of triatomic molecules. The molecules are assumed to be triangular, composed of massless rigid rods with atoms at the vertices.  The internal energy of a mole of the gas at temperature \(T\) is:

To raise the temperature of a certain mass of gas by \(50^\circ\text{C}\) at a constant pressure, \(160\) calories of heat is required. When the same mass of gas is cooled by \(100^\circ\text{C}\) at constant volume, \(240\) calories of heat is released. How many degrees of freedom does each molecule of this gas have (assume the gas to be ideal)?
1. \(2\)
2. \(5\)
3. \(6\)
4. \(3\)

Number of molecules in a volume of \(4~\text{cm}^3\) of a perfect monoatomic gas at some temperature \(T\) and at a pressure of \(2~\text{cm}\) of mercury is close to ? (Given, mean kinetic energy of a molecule (at \(T\)) is \(4 \times 10^{-14}\)erg, \(g=980\) cm/s² , density of mercury = \(13.6~ \text{g/cm}^3\))
1. \( 5.8 \times 10^{18} \)
2. \( 5.8 \times 10^{16} \)
3. \( 4.0 \times 10^{18} \)
4. \( 4.0 \times 10^{16}\)

Nitrogen gas is at a certain temperature \(300^\circ \text{C}.\) At what temperature (in Kelvin) will the root mean square (rms) speed of a hydrogen molecule be equal to the rms speed of a nitrogen molecule?
(given: molar mass of nitrogen molecule is \(28~\text g/ \text{mol}\) and molar mass of hydrogen molecule is \(2~\text g/ \text{mol}\))
1. \(21\) K
2. \(41\) K
3. \(52\) K
4. \(76\) K