Specific heat capacity of a substance is defined as the quantity of heat required to raise the temperature of 1 kg of the substance through 1K. Its unit is J kg?1K?1.

**Specific heat capacity**

*Specific heat capacity of a substance is defined as
the quantity of heat required to raise the temperature of 1 kg of the substance
through 1K. Its unit is J kg ^{?1}K^{?1}.*

*Molar specific heat capacity of a
gas*

*Molar specific heat capacity of a gas is defined as
the quantity of heat required to raise the temperature of 1 mole of the gas
through 1K. Its unit is J mol ^{?1} K^{?1}.*

Specific heat capacity of a gas
may have any value between ?∞ and +∞ depending upon the way in which heat energy is
given.

Let *m* be the mass of a gas and *C*
its specific heat capacity. Then ∆Q *= m ? C ?* ∆T where ∆Q is the amount of heat absorbed and ∆T is the corresponding rise in
temperature.

(i.e) C = ∆ Q / m ∆T

**Case (i)**

If the gas is
insulated from its surroundings and is suddenly compressed, it will be heated
up and there is rise in temperature, even though no heat is supplied from
outside

(i.e) ∆Q = 0

C=0

**Case (ii)**

If the gas is allowed
to expand slowly, in order to keep the temperature constant, an amount of heat
∆Q is supplied from outside,

Then C = ∆Q / (m x ∆T
) = ∆Q/0 = +infinity

(∵ ∆Q is +ve as heat is supplied from outside)

**Case (iii)**

If the gas is
compressed gradually and the heat generated ∆Q is conducted away so that
temperature remains constant, then

Then C = ∆Q / (m x ∆T
) = -∆Q/0 = -infinity

Thus we find that if the external
conditions are not controlled, the value of the specific heat capacity of a gas
may vary from +∞ to -∞

Hence, *in order to find the value of specific heat capacity of a gas,* *either the pressure or the volume of the gas
should be kept constant.*

Consequently a gas has two
specific heat capacities (i) Specific heat capacity at constant volume (ii)
Specific heat capacity at constant pressure.

*Molar specific heat capacity of a
gas at constant volume*

*Molar specific heat capacity of a gas at constant volume C _{V}
is defined as the quantity of heat required to raise the temperature of one
mole of a gas through 1 K, keeping its volume constant.*

*Molar specific heat capacity of a gas at constant pressure*

*Molar specific heat capacity of a gas at constant
pressure C _{p} is defined as the quantity of heat to raise the
temperature of one mole of a gas through 1 K keeping its pressure constant.*

*Specific heat capacity of
monoatomic, diatomic and triatomic gases*

Monoatomic gases like argon,
helium etc. have three degrees of freedom.

We know, kinetic energy per
molecule, per degree of freedom is ? kT.

Kinetic energy per molecule with
three degrees of freedom is 3/2 kT.

Total kinetic energy
of one mole of the monoatomic gas is given by E = 3/2. kT x N = 3/2 . RT, where
N is the Avogadro number.

dE/ dT = 3/2 R

If dE is a small
amount of heat required to raise the temperature of 1 mole of the gas at
constant volume, through a temperature dT,

dE = 1 ? C_{V}
? dT

C_{V} = dE/dT
= 3/2 . R

As R = 8.31 J mol^{?1}
K^{?1}

CV = 3/2 ? 8.31=12.465
J mol^{?1} K^{?1}

Then C_{P} ? C_{V}
= R

C_{P} = C_{V}
+ R

=3R /2 + R

= 5R/2 = (5/2) x 8.31

C_{p} = 20.775 J mol^{?1} K^{?1}

In diatomic gases like
hydrogen, oxygen, nitrogen etc., a molecule has five degrees of freedom. Hence
the total energy associated with one mole of diatomic gas is

E = 5 ? 1/2 kT ? N =
5/2 RT

Also, Cv = dE/dT =d/dT
(5/2 . RT) = 5/2 . R

C_{v} = ( 5/2
) . 8.31 = 20.775 J mol^{?1} K^{?1}

But C_{p} = C_{v}
+ R

=5/2 . R + R = (7/2)R

C_{p}
=(7/2) 8.31

= 29.085 J mol^{?1}
K^{?1}

similarly, Cp and Cv can be
calculated for triatomic gases.

*Internal energy*

*Internal energy U of a system is the energy possessed
by the system due to molecular motion and molecular configuration. The internal
kinetic energy U _{K} of the molecules is due to the molecular motion
and the internal potential energy U_{P} *is due to molecular
configuration. Thus

*U = U _{K} + U_{P}*

It depends only on the initial
and final states of the system. In case of an ideal gas, it is assumed that the
intermolecular forces are zero. Therefore, no work is done, although there is
change in the intermolecular distance. Thus *U _{P}*

In a real gas, intermolecular
forces are not zero. Therefore, a definite amount of work has to be done in
changing the distance between the molecules. Thus the internal energy of a real
gas is the sum of internal kinetic energy and internal potential energy. Hence,
it would depend upon both the temperature and the volume of the gas.

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11th 12th std standard Class Physics sciense Higher secondary school College Notes : Molar specific heat capacity of a gas |

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