What is the primary difference between specific heat capacity and molar heat capacity?

The primary difference lies in the basis of normalization. Specific heat capacity ($c$) refers to the heat required to raise the temperature of one unit mass (e.g., 1 gram or 1 kg) of a substance by one degree Celsius or Kelvin. Molar heat capacity ($C$), on the other hand, refers to the heat required to raise the temperature of one mole of a substance by one degree Celsius or Kelvin. Molar heat capacity is often more useful in chemical and physical contexts where the number of particles (moles) is a more relevant quantity than mass.

Why is $C_p$ always greater than $C_v$ for an ideal gas?

For an ideal gas, $C_p$ (molar heat capacity at constant pressure) is always greater than $C_v$ (molar heat capacity at constant volume) because of the work done during expansion. When heat is added at constant volume, all the energy goes into increasing the internal energy of the gas. However, when heat is added at constant pressure, the gas expands and does work on its surroundings. Therefore, more heat must be supplied at constant pressure to achieve the same temperature rise, as some of the energy is converted into mechanical work rather than solely increasing internal energy.

What are degrees of freedom, and how do they relate to molar heat capacity?

Degrees of freedom ($f$) represent the number of independent ways a molecule can store energy (translational, rotational, vibrational). According to the Law of Equipartition of Energy, each active degree of freedom contributes $\frac{1}{2}RT$ to the internal energy per mole. This directly impacts the molar heat capacity. For example, a monoatomic gas has 3 translational degrees of freedom, leading to $C_v = \frac{3}{2}R$. Diatomic gases typically have 5 degrees of freedom (3 translational + 2 rotational) at room temperature, resulting in $C_v = \frac{5}{2}R$.

What is Mayer's relation, and why is it important?

Mayer's relation states that for an ideal gas, the difference between molar heat capacity at constant pressure ($C_p$) and molar heat capacity at constant volume ($C_v$) is equal to the universal gas constant ($R$). Mathematically, $C_p - C_v = R$. This relation is crucial because it connects two fundamental thermodynamic properties and allows us to calculate one if the other is known, simplifying many thermodynamic calculations involving ideal gases.

Do solids and liquids have $C_p$ and $C_v$ values, and if so, how do they compare?

Yes, solids and liquids also have molar heat capacities at constant pressure ($C_p$) and constant volume ($C_v$). However, for solids and liquids, the difference between $C_p$ and $C_v$ is generally very small and often negligible. This is because solids and liquids expand very little upon heating compared to gases, meaning the work done against external pressure ($P\Delta V$) is minimal. Consequently, almost all the heat supplied goes into increasing their internal energy, whether at constant pressure or constant volume.

At what temperatures do vibrational degrees of freedom become active for gases?

Vibrational degrees of freedom typically become active at higher temperatures. At room temperature (around 300 K), the energy available is usually insufficient to excite the vibrational modes of most molecules. For diatomic gases like $O_2$ or $N_2$, vibrational modes might start contributing significantly to heat capacity above 1000 K. For NEET problems, unless explicitly stated or the temperature is very high, it's generally assumed that only translational and rotational degrees of freedom are active.

Molar Heat Capacities

Physics

NEET UG

Version 1Updated 22 Mar 2026

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Molar heat capacity, denoted by $C$ , is a fundamental thermodynamic property that quantifies the amount of heat energy required to raise the temperature of one mole of a substance by one degree Celsius (or one Kelvin). It is an intensive property, meaning it does not depend on the amount of substance, as it is normalized per mole. Unlike specific heat capacity which is per unit mass, molar heat ca…

Quick Summary

Molar heat capacity ( $C$ ) quantifies the heat required to raise the temperature of one mole of a substance by one Kelvin or Celsius. For gases, it's crucial to distinguish between molar heat capacity at constant volume ( $C_v$ ) and at constant pressure ( $C_p$ ).

$C_v$ represents the heat used solely to increase internal energy, while $C_p$ includes additional heat for work done during expansion. Mayer's relation, $C_p - C_v = R$ , links these two for ideal gases, with $R$ being the universal gas constant.

The values of $C_v$ and $C_p$ depend on the number of active degrees of freedom ( $f$ ) of the gas molecules (translational, rotational, vibrational), as per the Law of Equipartition of Energy. For monoatomic gases, $f=3$ , leading to $C_v = \frac{3}{2}R$ .

For diatomic gases at room temperature, $f=5$ , giving $C_v = \frac{5}{2}R$ . The ratio $\gamma = C_p/C_v$ is also a key parameter, related to $f$ by $\gamma = 1 + \frac{2}{f}$ .

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Key Concepts

Molar Heat Capacity at Constant Volume (

C_v

)

When a gas is heated in a rigid container (constant volume), it cannot expand, so no work is done against the…

Molar Heat Capacity at Constant Pressure (

C_p

)

When a gas is heated at constant pressure, it is allowed to expand. As it expands, it does work on its…

Degrees of Freedom and their contribution to Internal Energy

Degrees of freedom ( $f$ ) describe the number of independent ways a molecule can move or vibrate. For an ideal…

Molar Heat Capacity ($C$): — Heat for 1 mole,

1\text{ K}

temp rise. Units:

\text{J mol}^{-1}\text{ K}^{-1}

Constant Volume ($C_v$): —

C_v = \frac{1}{n} (\frac{dU}{dT})_v

. All heat increases internal energy. For ideal gas:

dU = n C_v dT

Constant Pressure ($C_p$): —

C_p = \frac{1}{n} (\frac{dQ}{dT})_p

. Heat increases internal energy AND does work.

Mayer's Relation: — For ideal gas,

C_p - C_v = R

Degrees of Freedom ($f$): — Monoatomic

f=3

, Diatomic

f=5

(at room temp), Polyatomic

f=6

(at room temp).

Equipartition Theorem: —

U = n \frac{f}{2} RT

$C_v$ from $f$: —

C_v = \frac{f}{2} R

$C_p$ from $f$: —

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

Ratio $\gamma$: —

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

Universal Gas Constant $R$: —

8.314 \text{ J mol}^{-1}\text{ K}^{-1}

To remember the degrees of freedom and $\gamma$ for common gases:

My Dog Plays 3 5 6 Rounds.

Monoatomic: 3 degrees of freedom (

f=3

Diatomic: 5 degrees of freedom (

f=5

Polyatomic: 6 degrees of freedom (

f=6

Remember

C_v = \frac{f}{2}R

and

\gamma = 1 + \frac{2}{f}

to quickly calculate the rest!