# Molar heat capacity of gases

### Heat capacity definition chemistry

The heat capacity of a substance is defined as the amount of heat required to raise the temperature by one degree.

Usually, the rise in temperature measured in centigrade and heat capacity expressed in calories. Heat capacity depends upon the amount of material in the system.

Heat capacity per gram of substance is called specific heat and per mole called molar heat capacity or simply heat capacity of the substances.

#### Heat capacity at constant volume and constant pressure

The magnitude 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.

The quantity of heat in the two operations would be different. Hence, in mentioning the heat capacity of gas, the condition has to specify.
1. Heat capacity at constant volume denoted by Cv.
2. Heat capacity at constant pressure denoted by Cp.

#### Molar heat capacity definition

The amount of heat required to raise the temperature of one gram of a substance by 1°K called specific heat.
The heat input required to rise by 1°K the temperature of one mole of the substances is called its molar heat capacity or simply heat capacity. The heat capacity denoted by C.

Molar heat capacity = molar mass × specific heat.

Cp = M × cp
Cv = M × cv
where Cp and Cv are the molar heat capacities at constant pressure and constant volume respectively. cp and cv are their specific heats.

Problem
The specific heat at constant pressure and constant volumes are 0.125 and 0.075 cal gm⁻¹ K⁻¹ respectively. Calculate the molecular weight and the atomicity of the gas. Name the gas if possible.

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

#### How is specific heat capacity measured?

Let dq heat required to increase the temperature dT for one mole of the substances and its heat capacity = C.
∴ C = dq/dT

dq is a path function and values of dq depend on the actual process followed. We can place certain restrictions to obtain precise values of heat capacity. The usual restrictions are at constant pressure and at constant volume.

Cv = [dq/dT]v and Cp = [dq/dT]p

Definition of internal energy and enthalpy,
dU = dqv and dH = dqp.

∴ Cv = [dq/dT]v = [dU/dT]v
Cp = [dq/dT]p = [dH/dT]p

#### Mechanical work in chemistry

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

In order to attain 1° 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 from mechanical work
Cp is some mechanical work is required as additional energy to absorb for lifting piston from volume V₁ to V₂.
Cp - Cv = Mechanical work = PdV
= P(V₂-V₁) = PV₂-PV₁.

If we consider the gas to be an ideal gas
PV = RT.

Cp - Cv = R(T+1) - RT
∴ Cp - Cv = R = 2 calories.

#### Determination of Cv from kinetic energy

Consider the monoatomic gases like argon, helium. If such gases heated at constant volume, the heat supplied will be entirely utilized for increasing the kinetic energy of the translation of the molecules.

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

Volume remaining constant, no energy can be used to do any mechanical work of expansion. 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, argon

Î” E = (3/2) [(T + 1) - T]R = (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 = Î”E = 3 calories.

For a mole of monoatomic gas the ratio of two heat capacities universally expressed by the symbol É£.

É£ = Cp/Cv = (Cv + R)/Cv
= (3 + 2)/3
= 1.66.

#### The total translational kinetic energy of gas molecules

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

Let x calories of heat used for increasing vibrational or rotational purposes.
∴ Cv = Î”E + x = (3/2)R + x.

Cp = Î”E + mechanical work + x
= (3/2)R + R + x
= (5/2)R + x.
Heat capacity difference remains the same, two calories, as monoatomic gas, but heat capacity ratio

É£ = {(3/2)R + x}/{(5/2)R + x}
= (5 + x)/(3 + x).

#### Heat capacity of gases from energy

Let molar energy of a gas E.
For monoatomic gas molecules

E = (3/2)RT
∴ Cv = [dE/dT]v = (3/2)R

For polyatomic linear gas molecules

E = (3/2)R + R + (3N - 5)R.
∴ Cv = [dE/dT]v = (3/2)R + R + (3N - 5)R

For polyatomic non-linear gas molecules

E = (3/2)R + R + (3N - 5)R.
∴ Cv = [dE/dT]v = (3/2)R + (3/2)R + (3N - 6)R

Study molar heat capacity of gases in chemistry, measurement of the heat capacity from mechanical work, kinetic energy, Cp - Cv for gas molecules

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