## Specific and Molar Heat Capacity

**Specific heat capacity** of gases is defined as the amount of heat required to raise the temperature of one gram gases by unit degree but per mole of gas is called molar heat capacity or simply **heat capacity**. Usually, heat capacity equation expressed at constant pressure (Cp) and volume (Cv) and energy units is used for its calculation.

### Heat capacity at constant pressure and volume

The amount of heat required to raise the temperature of one gram of a substance by 1^{0}K called specific heat and for one mole is called molar heat capacity.

C_{p} = M × c_{p
}C_{v} = M × c_{v}

where C_{p} and C_{v} are the heat capacity at constant pressure and constant volume respectively and c_{p} and c_{v} are their specific heat.

Problem

The specific heat capacity of gases at constant pressure and constant volumes are 0.125 and 0.075 cal gm^{-1} K^{-1} respectively, how to calculate the molecular weight and the gas formula from the specific heat equation. Name the gas if possible.

Solution

M = 40 and ⋎ = 1.66(mono-atomic), Ar(Argon).

### Calculation of specific heat capacity of gas

The calculation of heat capacity depends on the pressure and volume, especially in the cases of properties of gases.

- The temperature of a gm-mole of gas raised by one degree at constant volume.
- Again in another operation same raise of temperature allowing the volume to very.

Thus the observed quantity in the two operations would be different. Hence for mentioning this, the condition has to specify.

- Cp for gases at constant pressure.
- Cv for the gases at constant volume.

Cp and Cv of some gases at 1 atm pressure and 298 K temp are

Gas | C_{P} |
C_{V} |
γ |

Argon | 4.97 | 2.98 | 1.66 |

Helium | 4.97 | 2.98 | 1.66 |

Mercury | 5.00 | 3.00 | 1.67 |

Hydrogen | 6.85 | 4.86 | 1.40 |

Nitrogen | 6.96 | 4.97 | 1.40 |

Oxygen | 7.03 | 5.03 | 1.40 |

CO_{2} |
8.83 | 6.80 | 1.30 |

SO_{2} |
9.65 | 7.50 | 1.29 |

water | 8.67 | 6.47 | 1.34 |

Methane | 8.50 | 6.50 | 1.31 |

### Specific heat capacity equation

Let dq heat required to increase the temperature dT for one mole of the substances. From the thermodynamic equation for specific heat capacity

∴ C = dq/dT

where dq = path function

Values of dq depend on the actual process followed. But we can place certain restrictions to obtain precise values of C. The usual restrictions are at constant pressure and at constant volume.

C_{V} = (dq/dT)_{V} and C_{P} = (dq/dT)_{P}

From the thermodynamic definition of internal energy and enthalpy,

dU = (dq)_{V} and dH = (dq)_{P}

C_{V} = (dq/dT)_{V} = (dU/dT)_{V}

C_{P} = (dH/dT)_{P} = (dU/dT)_{P}

### Units of the molar and specific heat capacity

From the thermodynamics definition

C_{V} = (dq/dT)_{V} = (dU/dT)_{V}

C_{P} = (dH/dT)_{P} = (dU/dT)_{P}

∴ Specific heat units = unit of energy/unit of temperature.

Thus CGS and SI units of the heat capacity

erg K^{-1} and Joule K^{-1}

But if we maintained molar and specific heat capacity then per mole and per gram or kg used in these units.

### Cp – Cv from mechanical energy

A gas can be heated at constant pressure and constant volume in a cylinder fitted with a piston. The gas expands against the piston gives the mechanical energy.

In order to attain a 1^{0} rise in temperature, the heat supplied should be sufficient to provide the kinetic energy to the molecules and also able to do extra mechanical work.

Heat capacity Cp is some mechanical energy that is required to absorb for lifting the piston from volume V_{1} to V_{2}.

C_{P} – C_{V} = Mechanical energy = PdV

= P(V_{2} – V_{1})

= PV_{2} – PV_{1}

If we consider the gas to be an ideal gas

PV = RT

Thus C_{P} – C_{V} = R(T+1) – RT

∴ C_{P} – C_{V} = R = 2 calories

### Cp and Cv from the energy equation

Consider the monoatomic gases like argon or helium. If such gases are heated at constant volume, it utilized for increasing the kinetic energy of the translation.

Since the monoatomic gas molecules can not any absorption in vibrational or rotational motion.

If no heat used to do any mechanical work of expansion when the volume of the gas remains constant. Kinetic energy for one-mole gas at the temperature T denoted by E.

E = (3/2)PV = (3/2)RT

Increase of kinetic energy for 1° rise in temperature for monoatomic gas helium or argon

∴ ΔE = (3/2) × {R(T+1) – RT} = (3/2)R

=3 Calories

The heat supplied at a constant volume equal to the rise in kinetic energy per unit degree rise in temperature.

∴ Cv = ΔE = 3 calories

For a mole of monoatomic gas the ratio of Cp/Cv universally expressed by the symbol γ.

∴ γ = C_{P}/C_{V} = (C_{V} + R)/C_{V}

= (3 + 2)/3

= 1.66

#### Heat capacity of polyatomic gas

For polyatomic molecules, the haet supplied used up not only in increasing kinetic energy but also increasing vibrational or rotational energies.

Let x calories used for increasing vibrational or rotational purposes.

C_{P} – C_{V} = 2 calories remain constant for this energy equation but Cp/Cv calculation differing gas to gas.

### Calculation of Cp and Cv from energy equation

Heat supplied to one gm-mole of a gas kept at a constant volume to increase the temperature by one degree has the C_{V}.

### Calculation and experimental value of Cp and Cv

Experimental and calculation value of heat capacity Cp and Cv revels due to the following facts

Due to the perfect arrangement of the monoatomic gases

C_{V}/R = 1.5

Thus this value is independent of temperature over a wide range.

But for polyatomic gases, two points of disarrangement found for the observed heat capacities

- The calculation value of Cp and Cv always lower than the experimental values.
- They noticeably depend on the temperature.

CV/R = 3/2 + 3/2 +1 = 3.5 (calculated)

But the observed values of this lies in the range of 2.5 to 3.5.

Classical mechanics does not describe the variation of these molecular properties of polyatomic molecules. Thus for this purpose, we use quantum mechanics.

The principal of equipartition derived from the classical consideration of continuous absorption of energy governed by maxwell distribution.

But vibrational and rotational energy takes place in discrete quantities. Thus the observed value of specific heat capacity of gases explained only on the basis of quantum theory.