Degrees of Freedom and Molar Specific Heats MCQ Quiz - Objective Question with Answer for Degrees of Freedom and Molar Specific Heats - Download Free PDF

CONCEPT:

• The molar specific heat capacity of a gas at constant volume is defined as the amount of heat required to raise the temperature of 1 mol of the gas by 1 °C at the constant volume.

\({C_v} = {\left( {\frac{\Delta Q}{{n\Delta T}}} \right)_{constant\;volume}}\)

• The molar specific heat of a gas at constant pressure is defined as the amount of heat required to raise the temperature of 1 mol of the gas by 1 °C at the constant pressure.

\({C_p} = {\left( {\frac{{\Delta Q}}{{n\Delta T}}} \right)_{constant\;pressure}}\)

• The ratio of the two principal specific heat is represented by γ.

\(\therefore \gamma = \frac{{{C_p}}}{{{C_v}}}\)

• The value of γ depends on the atomicity of the gas.

EXPLANATION:

• The total internal energy of a mole of a rigid diatomic gas is

\(⇒ U=\frac{5}{2}RT\)

NOTE:

As we know, 

\(\Rightarrow C_v=\frac{dU}{dt}\)

\(\Rightarrow C_v=\frac{d}{dT}(\frac{5}{2}RT)=\frac{5}{2}R\)

Additional Information

As we know,

⇒ Cp - Cv = R

• Therefore, the molar specific heat of a gas at constant pressure is

\(\Rightarrow {C_p} = R+ {C_v}=R+\frac{5}{2}R=\frac{7}{2}R\)

Concept:

Degrees of Freedom (f):

The degrees of freedom of a molecule refer to the number of independent ways in which the molecule can move or store energy. For different types of gases:

• Monoatomic gas: f=3 (translational degrees of freedom)
• Diatomic gas: f=5 (3 translational + 2 rotational degrees of freedom)
• Polyatomic gas: f≥6 (includes translational, rotational, and vibrational degrees of freedom)

Mixture of Gases:

For a mixture of gases, the effective degrees of freedom f_eq
can be determined by the weighted average of the degrees of freedom of the individual gases:

\(\mathrm{f}_{\mathrm{eq}}=\frac{\mathrm{n}_1 \mathrm{f}_1+\mathrm{n}_2 \mathrm{f}_2}{\mathrm{n}_1+\mathrm{n}_2}\)
 
​Here,n and and are the number of moles and degrees of freedom of the gas, respectively.

Calculation:

\(\mathrm{f}_{\mathrm{eq}}=\frac{\mathrm{n}_1 \mathrm{f}_1+\mathrm{n}_2 \mathrm{f}_2}{\mathrm{n}_1+\mathrm{n}_2}\)

For diatomic gas f_eq = 5 

\(5=\frac{(\mathrm{N})(6)+(2)(3)}{\mathrm{N}+2}\)

5N + 10 = 6N + 6 

N = 4 

∴ The correct option is 3)

Concept: 

To find the molar-specific heat of the mixture at constant volume, we need to use the specific heat capacities of both monoatomic and diatomic gases.

For a monoatomic gas C_v = 3/2 R

For a diatomic gas Cv = 5/2 R

Calculation: 

C_V = \(\frac{\mathrm{n}_1 \mathrm{C}_{\mathrm{v}_1}+\mathrm{n}_2 \mathrm{C}_{\mathrm{v}_2}}{\mathrm{n}_1+\mathrm{n}_2}\)

⇒ CV= \(\frac{2 \times \frac{3}{2} R+6 \times \frac{5}{2} R}{2+6}\)

⇒ CV= \(\frac{9}{4}\)R

∴ The molar-specific heat of the mixture at a constant volume \(\frac{9}{4}R\)

Concept :

The molar specific heat capacity at constant volume (C_V) and at constant pressure (C_P) are related by the degrees of freedom of the gas molecules. For a diatomic gas, the degrees of freedom are 5, and for a monoatomic gas, the degrees of freedom are 3.

For a diatomic gas:

C_P = C

C_V (diatomic) = C - R, where R is the universal gas constant.

For a monoatomic gas:

C_P (monoatomic) = C_V (monoatomic) + R

C_V (monoatomic) = (3/2)R

C_P (monoatomic) = (5/2)R

Calculation:

The molar specific heat capacity of a diatomic gas at constant pressure is C.

Given C_P (diatomic) = C

We need to find C_V (monoatomic).

We know,

⇒ C_V (diatomic) = C - R

For a diatomic gas,

⇒ C = C_V (diatomic) + R

⇒ C = (5/2)R + R

⇒ C = (7/2)R

Now, for a monoatomic gas,

⇒ C_V (monoatomic) = (3/2)R

Using the relation:

⇒ C = (7/2)R

⇒ (3/2)R = (3/7)C

Hence,

∴ The molar specific heat capacity of a monoatomic gas at constant volume is (3C)/7.

Concept:

Degrees of Freedom:

• Degrees of freedom refer to the number of independent ways in which a system can move without violating any constraint.
• In the context of a diatomic molecule, we consider translational, rotational, and vibrational degrees of freedom.

Translational Degrees of Freedom:

• A diatomic molecule can move in 3-dimensional space, contributing 3 translational degrees of freedom.

Rotational Degrees of Freedom:

• A diatomic molecule can rotate about two perpendicular axes perpendicular to the bond axis, contributing 2 rotational degrees of freedom.
• Rotation around the bond axis is negligible due to the small moment of inertia.

Vibrational Degrees of Freedom:

• In the rigid rotator model, vibrational degrees of freedom are not considered.

Calculation:

Given,

Translational degrees of freedom = 3

Rotational degrees of freedom = 2

Total degrees of freedom for a diatomic molecule treated as a rigid rotator is,

⇒ Translational degrees of freedom + Rotational degrees of freedom

⇒ 3 + 2

⇒ 5

∴ The correct answer is 5.

CONCEPT:

Degrees of freedom:

• The degree of freedom (DOF) of a mechanical system is the number of independent parameters that define its configuration.
• The position and orientation of a rigid body in space are defined by three components of translation and three components of rotation, which means that it has six degrees of freedom..

EXPLANATION:

• The motion of a body as a whole from one point to another is called translation. 
• Thus, a molecule free to move in space has three translational degrees of freedom.
  • Molecules of monatomic gas have only translational degrees of freedom. 
• A monoatomic molecule free to move in space has 3 translational degrees of freedom.
• If a molecule is constrained to move along a line it requires one co-ordinate to locate it. Thus it has one degree of freedom for motion in a line.
• If a molecule is constrained to move in a plane it requires two coordinates to locate it. Thus it has two degrees of freedom for motion in a plane. Therefore option 3 is correct.                           

Concept:

• The molar heat capacity: It is defined as the amount of heat that is needed to raise the temperature of 1 mole of the substance through 1ºC. 

There are two types of molar specific heat:

• The molar specific heat capacity of a gas at constant volume: It is defined as the amount of heat required to raise the temperature of 1 mol of the gas by 1 °C at the constant volume.
• The molar specific heat of a gas at constant pressure: It is defined as the amount of heat required to raise the temperature of 1 mol of the gas by 1 °C at the constant pressure.
• If the specific heat is at constant pressure (C_P is greater than the specific heat at constant volume (C_V) then the gas constant can be expressed as CP - CV = R and this relation are termed as Mayer’s Formula.

Where, R = Gas constant, n = molar mass of the substance, CP= molar specific heat at constant pressure, CV = molar specific heat at constant volume.

Explanation:

As we have Mayer's formula;

CP - CV = R; C_P  = R + C_v....(i)

Also Cv = f/2 R...(where,f= degree of freedom = 3 for the monoatomic gases)

Hence, Equation (i) Becomes,

C_P = R + (3/2)R = (5/2)R

Key Points

• The degree of freedom of monoatomic gas is 3
• The degree of freedom of diatomic gas is 5
• The degree of freedom of polyatomic gas is 7

CONCEPT:

• The molar specific heat capacity of a gas at constant volume is defined as the amount of heat required to raise the temperature of 1 mol of the gas by 1 °C at the constant volume.

\({C_v} = {\left( {\frac{{\Delta Q}}{{n\Delta T}}} \right)_{constant\;volume}}\)

• The molar specific heat of a gas at constant pressure is defined as the amount of heat required to raise the temperature of 1 mol of the gas by 1 °C at the constant pressure.

\({C_p} = {\left( {\frac{{\Delta Q}}{{n\Delta T}}} \right)_{constant\;pressure}}\)

• The relation between the ratio of Cp and Cv with a degree of freedom is given by

\(\gamma = \frac{{{C_p}}}{{{C_v}}} = 1 + \frac{2}{f}\)

Where f = degree of freedom

EXPLANATION:

• The relation between the ratio of Cp and Cv with a degree of freedom is given by

\(\Rightarrow \gamma = \frac{{{C_p}}}{{{C_v}}} = 1 + \frac{2}{f}\)

• Diatomic gas has 5 degrees of freedom

\(\Rightarrow \gamma = 1 + \frac{2}{5} = \frac{5 +2}{5} = \frac{7}{5} \)

CONCEPT:

Degrees of freedom:

• The degree of freedom (DOF) of a mechanical system is the number of independent parameters that define its configuration.
• The position and orientation of a rigid body in space are defined by three components of translation and three components of rotation, which means that it has six degrees of freedom.
• It is given by N = 3A - R 

Where A = number of particles in the system and R = number of relations among the particles.

EXPLANATION:

• The motion of a body as a whole from one point to another is called translation. 
• The molecules of a diatomic gas like hydrogen, oxygen, nitrogen, etc has two atoms.
• Thus, a molecule of diatomic is free to move in space has three translational degrees of freedom and two rotational degrees of freedom.

For a diatomic gas,

The number of particle in the system (A) = 2

The number of relations among the particles (R) = 1

• The number of degrees of freedom

⇒ N = 3 × 2 -1 = 5

• Thus molecules of oxygen are free to move in space and have 5 translational degrees of freedom. Therefore option 3 is correct.                  

CONCEPT:

Degrees of freedom:

• The degree of freedom (DOF) of a mechanical system is the number of independent parameters that define its configuration.
• The position and orientation of a rigid body in space are defined by three components of translation and three components of rotation, which means that it has six degrees of freedom.
• It is given by N = 3A - R 

Where A = number of particles in the system and R = number of relations among the particles.

EXPLANATION:

• The motion of a body as a whole from one point to another is called translation. 
• Thus, a molecule free to move in space has three translational degrees of freedom.
  • Molecules of monatomic gas have only translational degrees of freedom.

For a monoatomic gas,

The number of particle in the system (A) = 1

The number of relations among the particles (R) = 0

• The number of degrees of freedom

⇒ N = 3A - R = 3 

• A monoatomic molecule free to move in space has 3 translational degrees of freedom.                                             

CONCEPT:

• The molar specific heat capacity of a gas at constant volume is defined as the amount of heat required to raise the temperature of 1 mol of the gas by 1 °C at the constant volume.

\({C_v} = {\left( {\frac{{\Delta Q}}{{n\Delta T}}} \right)_{constant\;volume}}\)

• The molar specific heat of a gas at constant pressure is defined as the amount of heat required to raise the temperature of 1 mol of the gas by 1 °C at the constant pressure.

\({C_p} = {\left( {\frac{{\Delta Q}}{{n\Delta T}}} \right)_{constant\;pressure}}\)

• The relation between the ratio of Cp and Cv with a degree of freedom is given by

\(\gamma = \frac{{{C_p}}}{{{C_v}}} = 1 + \frac{2}{f}\)

Where f = degree of freedom

EXPLANATION:

• The relation between the ratio of Cp and Cv with a degree of freedom is given by

\(\Rightarrow \gamma = \frac{{{C_p}}}{{{C_v}}} = 1 + \frac{2}{f}\)

Monoatomic gas has 3 degrees of freedom

\(\Rightarrow \gamma = 1 + \frac{2}{3} = \frac{3 +2}{3} = \frac{5}{3} \)

CONCEPT:

• Degree of Freedom (DOF): The number of independent ways by which a gas molecule can move, without any constraint imposed on it, is called the number of degrees of freedom.
  • For monoatomic molecule f = 3.
  • For diatomic molecule f = 5

CALCULATION:

• A diatomic molecule has a degree of freedom = 5, because
• It can move in translational motion in x y and z-direction.
• So the degree of freedom due to translational motion = 3
• Also due to rotational motion, degree of freedom = 2

So total degree of freedom = 3 + 2 = 5

• Hence the correct answer is option 3.

CONCEPT:

• Specific heat or specific heat capacity of a body is the amount of heat required for a unit mass of body to raise the temperature by 1 degree Celsius. It is denoted by C.
  • It is different for each substance.
• Specific heat capacity of Diatomic  gas:

​Total degree of freedom (f) = 3 rotational degree of freedom + 3 translational degree of freedom + 1 vibrational degree of freedom = 7

The relation between specific heat at constant pressure and at constant volume is given by:

CP – Cv = R

Cv = (f R)/2 and CP = (f + 2) R/ 2

Where R is universal gas constant

EXPLANATION:

For diatomic molecules: Degree of freedom (f) = 7

Specific heat capacity at constant pressure (CP) = (f + 2) R/ 2 = (7 + 2) R/ 2 = 9R/2.

So option 3 is correct.

CONCEPT:

• The molar specific heat capacity of a gas at constant volume is defined as the amount of heat required to raise the temperature of 1 mol of the gas by 1 °C at the constant volume.

\({C_v} = {\left( {\frac{\Delta Q}{{n\Delta T}}} \right)_{constant\;volume}}\)

• The molar specific heat of a gas at constant pressure is defined as the amount of heat required to raise the temperature of 1 mol of the gas by 1 °C at the constant pressure.

\({C_p} = {\left( {\frac{{\Delta Q}}{{n\Delta T}}} \right)_{constant\;pressure}}\)

• The ratio of the two principal specific heat is represented by γ.

\(\therefore \gamma = \frac{{{C_p}}}{{{C_v}}}\)

• The value of γ depends on the atomicity of the gas.

EXPLANATION:

• If a diatomic gas molecule is not a rigid rotator and has, in addition, one vibrational mode, then the total energy associated with a gram of diatomic gas molecules is

\(⇒ U=\frac{7}{2}RT\)

As we know, 

\(\Rightarrow C_v=\frac{dU}{dt}\)

\(\Rightarrow dU=\frac{d}{dT}(\frac{7}{2}RT)=\frac{7}{2}R\)

• The molar specific heat capacity of a gas at constant volume is \(C_v=\frac{7}{2}R\).

CONCEPT:

• The molar specific heat capacity of a gas at constant volume is defined as the amount of heat required to raise the temperature of 1 mol of the gas by 1 °C at the constant volume.

\({C_v} = {\left( {\frac{\Delta Q}{{n\Delta T}}} \right)_{constant\;volume}}\)

• The molar specific heat of a gas at constant pressure is defined as the amount of heat required to raise the temperature of 1 mol of the gas by 1 °C at the constant pressure.

\({C_p} = {\left( {\frac{{\Delta Q}}{{n\Delta T}}} \right)_{constant\;pressure}}\)

• The ratio of the two principal specific heat is represented by γ.

\(\therefore \gamma = \frac{{{C_p}}}{{{C_v}}}\)

• The value of γ depends on the atomicity of the gas.

EXPLANATION:

• A polyatomic gas molecule has, in general, three translation degree of freedom, three rotational degrees of freedom and certain number, say 'f' of vibrational mode, therefore the total energy of one mole of such gas is

⇒ U= (3 + f)RT

As we know, 

\(\Rightarrow C_v=\frac{dU}{dt}\)

\(\Rightarrow dU=\frac{d}{dT}(3+f)RT=(3+f)R\)

• The molar specific heat capacity of a gas at constant volume is C_v = (3 + f)R.