Degrees of Freedom and Molar Specific Heats MCQ Quiz in मल्याळम - Objective Question with Answer for Degrees of Freedom and Molar Specific Heats - സൗജന്യ PDF ഡൗൺലോഡ് ചെയ്യുക

CONCEPT:

Specific heat capacity of water:

• We can use the law of equipartition of energy to determine the specific heat of the water.
• We treat water like a solid.
• We consider every atom vibrating about its mean position. So the average energy of each atom is given as,

⇒ E = 3kBT

• The water molecule has three atoms, two hydrogens, and one oxygen.
• So the energy of one molecule of water is given as,

⇒ Ew = 9kBT

• If NA is the number of molecules in one mole of water, then the average energy of one mole of water is given as,

⇒ U = 9NAkBT = 9RT

Where kB = Boltzmann constant, T = absolute temperature, and R = universal gas constant

• So the molar specific heat of water is given as,

⇒ C = 9R

• This is the value observed and the agreement is very good.
• In the calorie, gram, degree units, water is defined to have unit-specific heat.
• The predicted specific heats are independent of temperature.

EXPLANATION:

• According to the law of equipartition of energy, the molar specific heat of water is given as,

⇒ U = 9RT

• So the molar specific heat of water is independent of mass and temperature. Hence, option 4 is correct.

CONCEPT:

Specific heat capacity of water:

• We can use the law of equipartition of energy to determine the specific heat of the water.
• We treat water like a solid.
• We consider every atom vibrating about its mean position. So the average energy of each atom is given as,

⇒ E = 3k_BT

• The water molecule has three atoms, two hydrogens, and one oxygen.
• If N_A is the number of molecules in one mole of water, then the average energy of one mole of water is given as,

⇒ U = 9N_Ak_BT = 9RT

Where k_B = Boltzmann constant, T = absolute temperature, and R = universal gas constant

• So the molar specific heat of water is given as,

⇒ C = 9R

• This is the value observed and the agreement is very good.
• In the calorie, gram, degree units, water is defined to have unit specific heat.
• The predicted specific heats are independent of temperature.

EXPLANATION:

• According to the law of equipartition of energy, the molar specific heat of water is given as,

⇒ C = 9R

Where R = universal gas constant

• Hence, option 1 is correct.

CONCEPT:

Specific heat capacity of solids:

• We can use the law of equipartition of energy to determine specific heats of solids.
• Consider a solid of N atoms, each vibrating about its mean position.
• We know that if a molecule oscillates in one dimension, then its average energy is given as,

⇒ E = kBT

• Since the molecule of a solid vibrates in three dimensions, so the average energy of the molecule of a solid is given as,

⇒ ES = 3kBT

• So the total energy in one mole of solid is given as,

⇒ U = 3NAkBT = 3RT

• Therefore the molar specific heat of the solid is given as,

\(\Rightarrow C=\frac{U}{T}=3R\)

Where R = universal gas constant

• The prediction generally agrees with experimental values at ordinary temperatures.
• As the molar specific heat of Carbon is 6.1 J/mol-K. So Carbon is an exception.

EXPLANATION:

• According to the law of equipartition of energy, the molar specific heat of the solid is given as,

\(\Rightarrow C=\frac{U}{T}=3R\)

• So the molar specific heat of solids is independent of temperature and mass. Hence, option 4 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.

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

Monatomic Gas:

• A monatomic gas is one in which atoms are not bound to each other.
• Monatomic gas consists of single atoms.
• Example: argon, krypton, and xenon.

​EXPLANATION:

• For a monoatomic gas, the molar heat capacity at constant volume is given as,

\(⇒ C_v = \dfrac{3}{2} R\)

• Hence, option 1 is correct.

Additional Information

• For a monoatomic gas, the molar heat capacity at constant pressure is given as,

\(⇒ C_P = \dfrac{5}{2} R\)

• For a monoatomic gas, the ratio of C_p and C_v is,

\(⇒ \frac{C_{p}}{C_{v}}=\frac{5}{2}\times\frac{2}{3}\)

\(⇒ \frac{C_{p}}{C_{v}}=\frac{5}{3}\)

\(⇒ \frac{C_{p}}{C_{v}}=1.67\)

The difference between Cp and Cv is given by Mayer's formula,

⇒ Cp - Cv = R

Calculation: 

Degree of freedom(f) = 5 + 2(3N – 5)

f = 5 + 2(3 × 2 –1) = 7

The energy of one molecule = \(\frac{\mathrm{f}}{2}\) K_BT

The energy of 10 molecules

⇒10 \(\left(\frac{\mathrm{f}}{2} \mathrm{~K}_{\mathrm{B}} \mathrm{T}\right)\) = 10 \(\left(\frac{7}{2} K_B T\right)\) = 35 KBT

∴ The Correct answer is Option (4): 35 KBT 

CONCEPT:

Specific heat capacity of water:

• We can use the law of equipartition of energy to determine the specific heat of the water.
• We treat water like a solid.
• We consider every atom vibrating about its mean position. So the average energy of each atom is given as,

⇒ E = 3kBT

• The water molecule has three atoms, two hydrogens, and one oxygen.
• If NA is the number of molecules in one mole of water, then the average energy of one mole of water is given as,

⇒ U = 9NAkBT = 9RT

Where kB = Boltzmann constant, T = absolute temperature, and R = universal gas constant

• So the molar specific heat of water is given as,

⇒ C = 9R

• This is the value observed and the agreement is very good.
• In the calorie, gram, degree units, water is defined to have unit specific heat.
• The predicted specific heats are independent of temperature.

CALCULATION:

Given N = 5 mole and ΔT = 10°C

• According to the law of equipartition of energy, the molar specific heat of water is given as,

⇒ C = 9R

Where R = universal gas constant

• So the amount of heat required to increase the temperature of N mole of water by ΔT temperature is given as,

⇒ E = NCΔT

⇒ E = 5 × 9R × 10

⇒ E = 450R

• Hence, option 3 is correct.

CONCEPT:

• Specific heat capacity at constant pressure (C_P): It is the amount of heat required to raise the temperature of 1 kg of gas maintained at constant pressure by 1 degree Celcius.
• Specific heat capacity at constant volume (CV): It is the amount of heat required to raise the temperature of 1 kg of gas maintained at constant volume by 1 degree Celcius.
• The SI unit for both C_P and C_V is J/(Kg.K).
• \(\gamma\) for a gas is the ratio of the specific heat capacity at constant pressure (C_p) to the specific heat capacity at constant volume (C_V). 

\(\Rightarrow \gamma = \frac{C_P}{C_V}\)

CALCULATION:

• For monoatomic gas,

 \(\gamma^m = \frac{C_P^m}{C_V^m}=\frac{5}{3} \\ C_P^m = \frac{5}{2} RT \\ C_V^m = \frac{3}{2} RT\)

In the above equations, the superscript m is just included to denote monoatomic gas. 

• For diatomic gas,

\(\gamma^d = \frac{C_P^d}{C_V^d}=\frac{7}{5} \\ C_P^d = \frac{7}{2} RT \\ C_V^d = \frac{5}{2} RT\)

In the above equations, the superscript d is just included to denote diatomic gas. 

• For a mixture of one mole of monoatomic gas and one mole of diatomic gas, that is a total of two moles of gas:

\(\Rightarrow C_P (net) = \frac{C_P^m +C_P^d}{1+1} = \frac{\frac{5}{2} RT + \frac{7}{2}RT}{2} = 3RT \\ \Rightarrow C_V (net) = \frac{C_V^m+C_V^d}{1+1}= \frac{\frac{3}{2} RT + \frac{5}{2}RT}{2} = 2RT \\ \Rightarrow \gamma (net) = \frac{C_P(net)}{C_V(net)} = \frac{3RT}{2RT} = 1.5\)

• Therefore, option 2 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:

• 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:

• 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:

Translational degrees of freedom refer to the independent movements a molecule can make in space.
Any molecule in 3D space can move along three perpendicular axes (x, y, z).
Rotational degrees of freedom refer to the independent ways a molecule can rotate around its center of mass.

Calculation:

Since CH₄ is polyatomic Non-Linear

The number of translational degrees of freedom for any molecule, including 
CH₄ = 3 

The number of rotational degrees of freedom for CH₄  = 3

∴ T he correct option is 2)