MHT-CET : Physics Entrance Exam

MHT - CET : Physics - Kinetic Theory of Gases Page 2

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Specific Heat of Gas

Specific Heat is defined as the quantity of heat required to raise the temperature of unit mass of a substance through 1°C (1 K).

If heat supplied to a substance = Q
Mass of substance =
m
Specific heat of substance = C
Rise in temperature of substance =
q

Q= mCq

 \ C= Q mq

 OR C= 1 dQ m dq

In case of gases, two specific heats are to be defined.
(i) Specific heat at constant volume
(ii) Specific heat at constant pressure
In case of isothermal expansion of a gas,

 q = 0 (No rise in temperature) Þ C = ¥

In case of adiabatic compression of a gas, its temperature increases but no heat enters or leaves the system.

\ Q= 0 Þ C = 0

Þ Specific heat of a gas can have any value from 0 to ¥ depending upon the conditions under which heat energy is supplied.

Hence, conveniently either the volume or pressure of a gas is kept constant to fix the value of its specific heat. Hence, two specific heats are defined.

The specific heats defined for unit mass are called principal specific heats.

Principal Specific Heat at Constant Volume (Cv)
The quantity of heat required to raise the temperature of unit mass of a gas through 1° when its volume is kept constant.

Principal Specific Heat at Constant Pressure (Cp)
The quantity of heat required to raise the temperature of unit mass of a gas through 1° when its pressure is kept constant.

 For principal specific heats Cp or Cv = 1 dQ m dT

SI unit = Joule/kg°K
Heat unit: kcal/kg°K

If a mole of a gas is considered instead of unit mass, the specific heats so defined are the molar specific heats.

Molar Specific Heat at Constant Volume (C'v): Is the quantity of heat required to raise the temperature of one mole of a gas through 1° when its volume is kept constant.

Molar Specific Heat at Constant Pressure (C'p): Is the quantity of heat required to raise the temperature of one mole of a gas through 1° when its pressure is kept constant.

 For molar specific heats C'p or C'v = 1 dQ n dT

SI unit = Joule/kg mole°K
Heat unit = kcal/kg mole°K

Relation between Molar and Principal Specific Heats
Let molecular weight of gas = M
Molar specific heat

 C' = 1 dQ = M dQ = M. 1 dQ =M.C n dT m dT m dT

\ C'p = M× Cp
and C'v = M× Cv
Molar specific heat = Molecular weight
´ Principal specific heat

To show that Cp > Cv

• When one mole (unit mass) of a gas is heated at constant volume, all the heat is used to raise its temperature.
• When one mole (unit mass) of the gas is heated at constant pressure, part of the heat is used to do external work while only the remaining heat is available for raising the temperature.

\ For the same rise of temperature,
Cp > Cv
i.e., Cp > Cv by an amount equal to the external work done.

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