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. \(13\)

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

Match the \(C_p/C_V\)  ratio for ideal gases with different type of molecules :
 

Given below are two statements :

In the light of the above statements, choose the correct answer from the options given below :
 

If the ratio of the number density per cm³ of the two gases is \(5:3\) and the ratio of the diameters of the molecules of the two gases is \(4:5,\) then, the ratio of the mean free path of molecules of two gases is: