## Definition of Specific and Molar Heat Capacity

**Heat capacity** (Specific) 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 (C_{p}) and volume (C_{v}) and energy unit is used for its calculation in physics or chemistry. Monoatomic gas molecules like helium, neon, argon when heated in constant volume, the heat supplied will be utilized in increasing the translational kinetic energy because these molecules have no vibrational or rotational moment. These mono-atomic gases at the constant volume, no energy can be used to do any mechanical work. But if we heated in constant pressure the gas expands against the piston and does mechanical work. For polyatomic gases, the supplied heat uses not only translational kinetic energy but also vibrational or rotational energies.

Solids also have heat capacities measured from Dulong petit experimental data that the atomic heat of all crystalline solid elements is the constant quantity and approximately equal to 6.4 calories. Atomic heat is the product of specific heat and the atomic weight of the element. This law holds for many periodic table elements like silver, gold, aluminum lead, iron, etc.

### Units of Specific and Molar Heat Capacity

Specific heat capacity is an extensive property with unit J K^{-1} kg^{-1} because the amount of heat required to raise temperature depends on the mass of the substances. But molar heat capacity is an intensive property in thermodynamics having the unit J K^{-1} mol^{-1}. We also use CGS and calories units to specify the heat capacities of the solid and gaseous substances. But if we maintained molar and specific heat capacity then per mole and per gram or kg used in these units.

### Heat Capacity at Constant Pressure and Volume

The amount of heat or thermal energy required to raise the temperature of one gram of a substance by 1°K is called specific heat and for one mole is called molar heat capacity. Therefore, C_{p} = M × c_{p} and C_{v} = M × c_{v}, where C_{p} and C_{v} are measured at constant pressure and constant volume respectively, and c_{p} and c_{v} are their specific heat. The calculation of C_{p} or C_{v} 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 is called heat capacity at constant volume or simply C_{v}. Again in another operation same raise of temperature done at constant pressure is called heat capacity at constant pressure or simply C_{p}.

Therefore, the observed quantity in the two operations would be different. Hence for measuring heat capacity, the condition of pressure and volume must have to specify. In learning chemistry and physics, C_{p}, C_{v,} and C_{p}/C_{v} or γ of some gases at 1 atm pressure and 298 K temperature are given below the table,

Gases | 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 |

Problem: The C_{p} and C_{v} of gases 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).

### C_{p} – C_{v} from Mechanical Work or Energy

A monoatomic gas can be heated at constant pressure and constant volume in a cylinder fitted with a piston. When the gas expands against the piston gives mechanical energy. In order to attain a 1° rise in temperature, the heat supplied should be sufficient to increase the energy of the molecules and also able to do extra mechanical work. Therefore, C_{p} equal to some mechanical energy required for lifting the piston from volume V_{1} to V_{2}. C_{P} – C_{V} = mechanical work or energy = PdV = P(V_{2} – V_{1}) = PV_{2} – PV_{1}. If the gases obey ideal gas law, PV = RT. Therefore, C_{p} – C_{v} = R(T + 1) – RT, or C_{p} – C_{v} = R = 2 calories.

### Heat Capacity of Monoatomic Gases

Consider the monoatomic gases like argon or helium. If such gases are heated at constant volume, it is 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 is used to do any mechanical work of expansion when the volume of the gas remains constant. Therefore, the Kinetic energy for one-mole ideal gases at T temperature, E = 3PV/2 = 3RT/2. Increase of kinetic energy for 1° rise in temperature for monoatomic gas helium or argon, ΔE = 3{R(T+1) – RT}/2 = 3R/2 =3 calories.

The heat supplied at a constant volume equal to the rise in kinetic energy per unit degree rise in temperature. Therefore, C_{v} = ΔE = 3 calories. For one mole of monoatomic gas, the ratio of C_{p}/C_{v} universally expressed by the symbol γ calculated by the following equation, γ = C_{p}/C_{v} and C_{p} – C_{v} = R. Therefore, γ = (C_{v} + R)/C_{v}

= (3 + 2)/3 = 1.66.

### Thermodynamic Definition of Heat Capacity

Hence like internal energy, enthalpy, entropy, and free energy heat capacity also a thermodynamic property. Let dq energy required to increase the temperature dT for one mole of the gaseous substances. Therefore, the thermodynamic definition of the specific heat capacity, C = dq/dT, where dq = path function. Hence the values of heat change depend on the actual process which followed for this measurement. But we can place certain restrictions to obtain precise values of C_{p} and C_{v}. The usual restrictions are at constant pressure and at constant volume. Therefore, C_{v} = (dq/dT)_{v} = (dU/dT)_{v} and C_{p} = (dH/dT)_{p} = (dU/dT)_{p}.

### Polyatomic Gas Molecule

For polyatomic molecules, the heat supplied used up not only in increasing kinetic energy but also in increasing vibrational or rotational energies. Let x calories be used for increasing vibrational or rotational purposes.

C_{p} – C_{v} = 2 calories remain constant for this energy equation but C_{p}/C_{v} calculation differing gas to gas.

### Calculation of C_{v} from Energy Equation

Heat supplied to one gram-mole of a gas kept at a constant volume to increase the temperature by one degree has the C_{v} for monoatomic or polyatomic gases. But monoatomic gases use this energy to increase the translational kinetic energy and polyatomic gases use it to increases translational, vibrational, and rotational kinetic energy.

### Calculation and experimental value of Cp and Cv

Experimental and calculation values of C_{p} and C_{v} revels due to the following facts. Due to the perfect arrangement of the monoatomic gases, C_{v}/R = 1.5. Therefore, the value of C_{p} and C_{v} independent of temperature over a wide range. For polyatomic gases, two points of disarrangement found, the first always lower than the predicted value and the second is noticeable dependent on temperature.

The observed values of heat capacity for polyatomic gases lie between the range of 2.5 to 3.5. Classical mechanics does not describe the variation of these molecular properties. Therefore, we use quantum mechanics. The principle of equipartition is derived from the classical consideration of continuous energy absorption by atom governed by Maxwell distribution. Vibrational and rotational energy take place in discrete units, but the measured value of specific heat capacity of gas or gases is explained only on the basis of the quantum equation. At high temperatures, the energy levels are quite close and the observed spectrum would be continuous and the heat capacity equation for gas molecule would be valid.