For a mass $m$ of an ideal gas at constant pressure $P$,the volume versus temperature graph is represented by the straight line $B$. If the mass is changed to $2m$ and the pressure is changed to $2P$,which straight line will represent the new state?

$A$ piston is slowly pushed into a metal cylinder containing an ideal gas. Which of the following statements is/are incorrect?

Consider the following statements for air molecules in an air-tight container:
$(I)$ The average speed of molecules is larger than root mean square speed.
$(II)$ Mean free path of molecules is larger than the mean distance between molecules.
$(III)$ Mean free path of molecules increases with temperature.
$(IV)$ The rms speed of nitrogen is smaller than oxygen molecule.
Which of the above statements are correct?

Assertion $(A):$ The total translational kinetic energy of all the molecules of a given mass of an ideal gas is $1.5$ times the product of its pressure and volume.
Reason $(R):$ The molecules of gas collide with each other and the velocities of the molecules change due to the collision.

Three vessels of equal capacity have gases at the same temperature and pressure. The first vessel contains neon (monatomic),the second contains chlorine (diatomic),and the third contains uranium hexafluoride (polyatomic). Do the vessels contain equal number of respective molecules? Is the root mean square speed of molecules the same in the three cases? If not,in which case is $v_{rms}$ the largest?

Two rigid boxes containing different ideal gases are placed on a table. Box $A$ contains one mole of nitrogen at temperature $T_0$,while Box $B$ contains one mole of helium at temperature $(7/3)T_0$. The boxes are then put into thermal contact with each other,and heat flows between them until the gases reach a common final temperature (ignore the heat capacity of the boxes). The final temperature of the gases,$T_f$,in terms of $T_0$ is: