Past Year Questions: Theory of Gases | Physical Chemistry PDF Download

Naphthalene (C₁₀H₈) has 10 carbon atoms and 8 hydrogen atoms.

1. Calculate the total number of atoms:

   • Naphthalene has a total of 10 (carbon) + 8 (hydrogen) = 18 atoms.

2. Degrees of freedom:

   • Each atom in a molecule has 3 degrees of freedom (x, y, z directions).
   • Therefore, naphthalene has 18×3=54 degrees of freedom.

3. Translation and rotation degrees of freedom:

   • For a non-linear molecule, there are 3 translational and 3 rotational degrees of freedom.
   • Thus, the total number of vibrational modes is obtained by subtracting the translational and rotational degrees of freedom from the total degrees of freedom.

Normal modes of vibration=Total degrees of freedom−Translational degrees of freedom−Rotational degrees of freedom

Normal modes of vibration=54−3−3=48.

From the kinetic theory of gases, the mathematical expression of the most probable speed and the root mean square speed are : C_mp = √[2RT/M] and C_rms = V[3RT/M]

where, R = Universal gas constant, T = Absolute temperature and M = Molar mass of gas 

C_mp/C_rms = √2√3

Hence, the correct answer is : B.

In the van der Waals equation, b term stands for the volume correction term. Its value is given by : b = 4 N_A (4/3) πr³ ; where r = radius of the gas molecule From the equation, it is evident that as the value of r increase, b value will also increase. As O atom is smaller than N atom, 0 2 has smaller r value than N₂. Hence, b value will be smaller for O₂ than N₂.
Hence, the correct answer is : C.

Acetylene (C2H2) is a tetra-atomic molecule. Hence, the total degree of freedom = 12. For linear tetra-atomic molecule, the translational, rotational and vibrational degree of freedoms are 3, 2 and 7 respectively. The translational, rotational and vibrational degree of freedom will contribute 1/2 kT, 1/2 kT and kT.
The translational kinetic energy, ∈ = 3/2 kT ; The rotational kinetic energy, ∈ = kT ; The vibrational kinetic energy, ∈ = 7 kT Hence, the total kinetic energy = 9.5 kT Hence, the molar kinetic energy, E = 9.5 NkT = 9.5 RT
Hence, the molar heat capacity at constant volume, C_v,m = d/dT[9.5 RT] = 9.5R

Hence, the correct answer is : D.

The major force present in Ar, NH₄Cl, HF and HCl are London or, van der Walls force, ionic force, H-bonding as well as Debye force and Debye forces respectively. The order of strength of intermolecular force runs as : Ionic force > H-bond > Debye force > London force.
Hence, the strength interatomic/intermolecular forces follows the order : 2 > 3 > 4 > 1.
Hence, the correct answer is : A.

CO₂ is a linear tri-atomic molecule while SO₂ and H₂O are non-linear tri-atomic molecules.
In case of CO₂: It is a linear tri-atomic molecule. The total degrees of freedom = 3N = 9. The translational, rotational and vibrational degrees o f freedom for C 0 2 molecule are 3, 2 and 4 respectively. Hence, the total molar energy of the system, E_molar = 3/2 RT + 2/2 RT + 4 R T 13/2 RT.
Hence, the molar heat capacity at constant volume, C_v = (dE/dT)_v =13/2 R = 6.5 R.

In case of SO₂ and H₂O : These are non-linear tri- atomic molecules. The total degrees of freedom = 3N = 9. The translational, rotational and vibrational degrees of freedom for these molecule are 3, 3 and 3 respectively. Hence, the total molar energy of the system, E_molar =3/2 RT + 3/2 RT + 3RT = 6 RT.
Hence, the molar heat capacity at constant volume, C_v - (dE/dT)_v = 6 R.

Hence, the molar heat capacity of CO₂ at constant volume is greater than the molar heat capacity of H₂O and SO₂ at constant volume.
Hence, the correct answer is : B.

The mathematical expression for the average speed is as follows :

Hence, from the expression of C_av, it is evident that higher is the molecular weight of the gas molecule, lower will be the value of C_av.

The molecular weight of the following gas molecules is as follows : O₂ > N₂ > H₂.

Hence, the average speed of the given gas molecules is as follow's : H₂ > N₂ > O₂.

Hence, the correct answer is : A.

The total energy of a molecule is the sum of its translational, rotational and vibrational energy. Carbon dioxide is a linear tri-atomic molecule. The translational, rotational and vibrational degrees of freedom of CO₂ are 3, 2 and (9 - 5) = 4 respectively. Hence, the translational, rotational and vibrational energies of CO₂ molecule are .3/2RT, RT and 4RT. Hence, applying the principle of equipartition of energy, the molar heat capacity of CO₂ at constant volume C_v.m = d/dT(3/2RT + RT + 4 RT) = d/dT(6.5RT) = 6.5R.

Hence, the correct answer is : C.

In case o f linear and non-linear molecules, there are three translational degrees of freedom, each of which will contribute 1/2 RT. Hence, the total contribution towards internal energy is 3/2 RT.
In case of linear' molecule, there are two rotational degrees of freedom, each of which will contribute 1/2 RT. Hence, the total contribution towards internal energy is RT.
In case of non-linear molecule, there are three rotational degrees of freedom, each of which will contribute 1/2 RT. Hence, the total contribution towards internal energy is 3/2 RT.
Hence, in case of linear molecule, the total contribution towards internal energy by translational and rotational degrees of freedom : (3/2 + 1)RT = 5/2 RT.

Hence, in case of linear molecule the molar internal energies of the gas at temperature T, U_m(T) and at T = 0., U_m(0) are related as : U_m(T) = U_m(0) + 5/2RT (Considering translation and rotation only).
Hence, Ihe correct answer is : B.

The expression o f root-meati-square velocity is V_rms = (3 RT/M)^(1/2)

According to the question,

Hence, the ratio T(H₂)/T(He) has the value 1/2. Hence, the correct answer is : A.

The equation of state for an ideal gas is as follows : PV = nRT

Where, P = pressure of the gas ; V = volume of the gas ; n = no. of moles of the gas

T = absolute temperature of the gas ; R = Universal gas constant.

From this relation one can easily written PV = K = Constant

Hence, a plot of P vs. V will be a rectangular parabola which is non-iinear in nature.

Hence, the correct answer is : C.

We know that the most probable speed is given by :

C_mp = (2RT/M)^(1/2) =1000

or, 2RT/M= 1000000

or, T = 500000M/R

The average speed is given by

C_av = (8RT/πM)^(1/2) = {(8R X 500000M)/(RMπ)}^(1/2)

=1128 ms^-1

Hence, the correct answer is : B.

- The Maxwell-Boltzmann distribution describes how the speeds of molecules in a gas vary with temperature.
- As temperature decreases:
- H (Height): Increases. Lower temperature means more molecules have speeds close to the average, increasing peak height.
- L (Location of Maximum): Decreases. Lower temperature means the most probable speed decreases.
- W (Width at Half Height): Decreases. Lower temperature leads to a narrower distribution of speeds.
- Therefore, option C is correct: H decreases, L decreases, and W decreases.

From the above diagram, it is evident that as the temperature is increased, broadening of the curve occurs. Hence, the curve maximum shift towards right and it become flatten. Hence, the height of the peak decreases while the location of the peak increases and also width at the half height increases.
Hence, the correct answer is : C.

In case of N₂ : It is a linear, homo-nuclear diatomic molecule. Hence, the total degrees of freedom = 3.2 = 6.

The translational, rotational and vibrational degrees of freedoms are 3, 2 and 1 respectively. Hence, the E_trans = 3/2 RT ; E_rot = RT and E_vib = 0 at room temperature.

Hence, the heat capacity at constant volume, C_v = d/dT [(3/2 + 1 )RT] = 5/2R = 20.785 J/mol/K In case of O₂ : It is a linear, homo-nuclear diatomic molecule. Hence, the total degrees of freedom = 3.2 = 6.

The translational, rotational and vibrational degrees of freedoms are 3, 2 and 1 respectively. Hence, the E_trans = 3/2 RT ; E_rot = RT and E_Vib = 0 at room temperature.

Hence, the heat capacity at constant volume, C_v = d/dT [(3/2 + 1)RT] = 5/2R = 20.785 J/mol/K

In case of CO₂ : It is a linear, hetero-nuclear triatomic molecule. Hence, the total degrees of freedom = 3.3=9.

The translational, rotational and vibrational degrees of freedoms are 3, 2 and 4 respectively. Hence, the E_trans = 3/2 RT ; E_rot = RT and E_Vib = 0 at room temperature.

Hence, the heat capacity at constant volume, C_v = d/dT [(3/2 + 1)RT] = 5/2R = 20.785 J/mol/K

In case  of SO₂ : It is a non-linear, hetero-nuclear triatomic molecule. Hence, the total degrees of freedom = 3.3 = 9.

The translational, rotational and vibrational degrees of freedoms are 3, 3 and 3 respectively. Hence, the E_trans = 3/2 RT ; E_rot = 3/2 RT and E_vib = 0 at room temperature.

Hence, the heat capacity at constant volume, C_v = d/dT [(3/2 + 3/2)RT] = 3R = 24.942 J/mol/K = 25 J/mol/K.

Hence, the colourless gas is SO₂.
Hence, the correct answer is : D.

The most probable velocity (v_mp), average velocity (v_av) and root mean square velocity (v_rms) are defined as :

  From these equations it is clear that the value of root the mean square velocity is highest while the value of the most probable velocity is lowest. Hence, the decreasing order of various types of velocities : v_rms > V_av > V_mp. 

Hence, the correct answer is : A.