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1.

A given sample of an ideal gas occupies a volume \(V\) at a pressure \(P\) and absolute temperature \(T\). The mass of each molecule of the gas is \(m\). Which of the following gives the density of the gas?

1.	\(\dfrac{P}{kT}\)	2.	\(\dfrac{Pm}{kT}\)
3.	\(\dfrac{P}{kTV}\)	4.	\(mkT\)

2.

A gas mixture consists of \(2\) moles of \(\mathrm{O_2}\) and \(4\) moles of \(\mathrm{Ar}\) at temperature \(T.\) Neglecting all the vibrational modes, the total internal energy of the system is:

1.	\(15RT\)	2.	\(9RT\)
3.	\(11RT\)	4.	\(4RT\)

3.

4. Two vessels separately contain two ideal gases \(A\) and \(B\) at the same temperature, the pressure of \(A\) being twice that of \(B.\) Under such conditions, the density of \(A\) is found to be \(1.5\) times the density of \(B.\) The ratio of molecular weight of \(A\) and \(B\) is:

1.	\(\dfrac{2}{3}\)	2.	\(\dfrac{3}{4}\)
3.	\(2\)	4.	\(\dfrac{1}{2}\)

5.

One mole of an ideal diatomic gas undergoes a transition from \(A\) to \(B\) along a path \(AB\) as shown in the figure. 
         
The change in internal energy of the gas during the transition is:

6. The ratio of the specific heats \(\frac{{C}_{{P}}}{{C}_{{V}}}=\gamma\)  in terms of degrees of freedom \((n)\) is given by:

1.	\(\left(1+\frac{1}{n}\right )\)	2.	\(\left(1+\frac{n}{3}\right)\)
3.	\(\left(1+\frac{2}{n}\right)\)	4.	\(\left(1+\frac{n}{2}\right)\)​

7.

The mean free path of molecules of a gas (radius \(r\)) is inversely proportional to:

1.	\(r^3\)	2.	\(r^2\)
3.	\(r\)	4.	\(\sqrt{r}\)

8.

In the given \({(V\text{-}T)}\) diagram, what is the relation between pressure \({P_1}\) and \({P_2}\)? 

1.	\(P_2>P_1\)	2.	\(P_2<P_1\)
3.	cannot be predicted	4.	\(P_2=P_1\)

9.

The amount of heat energy required to raise the temperature of \(1\) g of Helium at NTP, from \({T_1}\) K to \({T_2}\) K is:

1.	\(\dfrac{3}{2}N_ak_B(T_2-T_1)\)	2.	\(\dfrac{3}{4}N_ak_B(T_2-T_1)\)
3.	\(\dfrac{3}{4}N_ak_B\frac{T_2}{T_1}\)	4.	\(\dfrac{3}{8}N_ak_B(T_2-T_1)\)

10.

At \(10^{\circ}\text{C}\) the value of the density of a fixed mass of an ideal gas divided by its pressure is \(x.\) At \(110^{\circ}\text{C}\) this ratio is:

1.	\(x\)	2.	\(\dfrac{383}{283}x\)
3.	\(\dfrac{10}{110}x\)	4.	\(\dfrac{283}{383}x\)

11.

An increase in the temperature of a gas-filled container would lead to:

1.	decrease in intermolecular distance.
2.	increase in its mass.
3.	increase in its kinetic energy.
4.	decrease in its pressure.

12.

The value \(\gamma = \frac{C_P}{C_V}\) for hydrogen, helium, and another ideal diatomic gas \(X\) (whose molecules are not rigid but have an additional vibrational mode), are respectively equal to:

1.	\(\dfrac{7}{5}, \dfrac{5}{3}, \dfrac{9}{7}\)	2.	\(\dfrac{5}{3}, \dfrac{7}{5}, \dfrac{9}{7}\)
3.	\(\dfrac{5}{3}, \dfrac{7}{5}, \dfrac{7}{5}\)	4.	\(\dfrac{7}{5}, \dfrac{5}{3}, \dfrac{7}{5}\)

13.

The mean free path for a gas, with molecular diameter \(d\) and number density \(n,\) can be expressed as:

1.	\( \dfrac{1}{\sqrt{2} n \pi {d}^2} \)	2.	\( \dfrac{1}{\sqrt{2} n^2 \pi {d}^2} \)
3.	\(\dfrac{1}{\sqrt{2} n^2 \pi^2 d^2} \)	4.	\( \dfrac{1}{\sqrt{2} n \pi {d}}\)

14.

A cylinder contains hydrogen gas at a pressure of \(249~\text{kPa}\) and temperature \(27^\circ\text{C}.\) Its density is:
(\(R=8.3~\text{J mol}^{-1} \text {K}^{-1}\))
1. \(0.2~\text{kg/m}^{3}\)
2. \(0.1~\text{kg/m}^{3}\)
3. \(0.02~\text{kg/m}^{3}\)
4. \(0.5~\text{kg/m}^{3}\)

15. The average thermal energy for a mono-atomic gas is:
(\(k_B\) is Boltzmann constant and \(T\) absolute temperature)

1.	\(\dfrac{3}{2}k_BT\)	2.	\(\dfrac{5}{2}k_BT\)
3.	\(\dfrac{7}{2}k_BT\)	4.	\(\dfrac{1}{2}k_BT\)

16.

The mean free path \(l\) for a gas molecule depends upon the diameter, \(d\) of the molecule as:

1.	\(l\propto \dfrac{1}{d^2}\)	2.	\(l\propto d\)
3.	\(l\propto d^2 \)	4.	\(l\propto \dfrac{1}{d}\)

17. An ideal gas equation can be written as \(P = \dfrac{ρRT}{M_{0}}\) where \(\rho\) and \(M_{0}\) are respectively:

1.	mass density, the mass of the gas.
2.	number density, molar mass.
3.	mass density, molar mass.
4.	number density, the mass of the gas.

18.

If \(C_P\) and \(C_V\) denote the specific heats (per unit mass) of an ideal gas of molecular weight \(M\) (where \(R\) is the molar gas constant), the correct relation is:
1. \(C_P-C_V=R\)
2. \(C_P-C_V=\frac{R}{M}\)
3. \(C_P-C_V=MR\)
4. \(C_P-C_V=\frac{R}{M^2}\)

19.

To find out the degree of freedom, the correct expression is:
1. \(f=\frac{2}{\gamma -1}\)
2. \(f=\frac{\gamma+1}{2}\)
3. \(f=\frac{2}{\gamma +1}\)
4. \(f=\frac{1}{\gamma +1}\)

20.

The equation of state for 5g of oxygen at a pressure P and temperature T, when occupying a volume V, will be: (where R is the gas constant)
1. PV = 5 RT
2. PV = (5/2) RT
3. PV = (5/16) RT
4. PV = (5/32) RT

21.

Match Column-I and Column-II and choose the correct match from the given choices.

	Column-I		Column-II
(A)	Root mean square speed of gas molecules	(P)	\(\dfrac13nm\bar v^2\)
(B)	The pressure exerted by an ideal gas	(Q)	\( \sqrt{\dfrac{3 R T}{M}} \)
(C)	The average kinetic energy of a molecule	(R)	\( \dfrac{5}{2} R T \)
(D)	The total internal energy of a mole of a diatomic gas	(S)	\(\dfrac32k_BT\)

 

	(A)	(B)	(C)	(D)
1.	(Q)	(P)	(S)	(R)
2.	(R)	(Q)	(P)	(S)
3.	(R)	(P)	(S)	(Q)
4.	(Q)	(R)	(S)	(P)

22. The volume occupied by the molecules contained in \(4.5~\text{kg}\) water at STP, if the molecular forces vanish away, is:

1.	\(5.6~\text{m}^3\)	2.	\(5.6\times10^{6}~\text{m}^3\)
3.	\(5.6\times10^{3}~\text{m}^3\)	4.	\(5.6\times10^{-3}~\text{m}^3\)

23.

The temperature at which the RMS speed of atoms in neon gas is equal to the RMS speed of hydrogen molecules at \(15^{\circ} \text{C}\) is:
(the atomic mass of neon \(=20.2~\text u,\) molecular mass of hydrogen \(=2~\text u\))
1. \(2.9\times10^{3}~\text K\)
2. \(2.9~\text K\)
3. \(0.15\times10^{3}~\text K\)
4. \(0.29\times10^{3}~\text K\)

24. Three vessels of equal capacity have gases at the same temperature and pressure. The first vessel contains helium (monoatomic), the second contains fluorine (diatomic) and the third contains sulfur hexafluoride (polyatomic). The correct statement, among the following, is:

1.	All vessels contain an unequal number of respective molecules.
2.	The root mean square speed of molecules is the same in all three cases.
3.	The root mean square speed of helium is the largest.
4.	The root mean square speed of sulfur hexafluoride is the largest.