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# 7. State first law of thermodynamics on its basis establish the relation betweentwo molar specific heat for a gas.

## 1 Answers

3 years ago

Dear Vaibhav

The first law of thermodynamics is a version of the law of conservation of energy, adapted for thermodynamics systems. The law of conservation of energy states that the total energy of an isolated system is constant; energy can be transformed from one form to another, but can be neither created nor destroyed.

Consider one mole of a perfect gas enclosed in a perfectly conducting cylinder. Let the cylinder be fitted with a perfectly conducting, frictionless piston.

Keeping the volume constant, let the temperature of the gas be increased from T to T + dT. If Cv is the specific heat of the gas at constant volume, then the heat supplied to increase the internal energy of the gas is CvdT.

Suppose that now we supply heat energy to the gas and allow the piston to move so as to keep the pressure constant. If the temperature changes from T to T + dT, then the part of the total energy supplied that is used in increasing the internal energy of the gas is equal to CydT. If the piston moves through a distance dx to keep the pressure constant, then the amount of external work done by the heat energy is

W = Force x Displacement

= Pressure x Surface area of the piston x Displacement = (P) (A) (dx) = P dV where dV = A dx is the increase in the volume of the gas.

Thus,

f Total heat supplied 1 = /Heat supplied at j + External work done i at constant pressure J Iconstant volume J

i.e., Cp dT = Cv dT + PdV      ....... (1)

For a perfect gas,

PV = RT

.-. P (V + dV) = R (T+ d T) PV + PdV = RT + R dT

P dV = R dT, since PV = RT Hence eq (1) becomes

Cp dT = Cv dT + R dT

or Cp = Cv + R

or Cp- Cv = R

This relation is known as Mayer's equation.

Note : For any mass m of the gas, cp - cy = r = nR where n = is the number of moles present in mass m.

Regards

Arun (askIITians forum expert)

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