CBSE Class 11-science Answered

Define the specific heat of a gas at constant pressure. Derive the relation between cp and cv.

Asked by seeni2005 | 11 Jan, 2021, 09:45: PM

Expert Answer

Specific heat at constant pressure is defined as the quantity of heat required to raise the temperature of unit mass

of gas by 1 ^oC at constant pressure .

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From first law of thermodynamics , we have , ΔQ = ΔU + ΔW ................ (1)

where ΔQ is the quatity of heat absorbed by gas , ΔU is change in internal energy of the gas and

ΔW is workdone by the gas . Let us assume all the quantities given in eqn.(1) are for one mole of gas .

Workdone by the gas is wriiten as , ΔW = P ΔV ..................(2)

where P is pressure and ΔV is change in volume . Using eqn.(2) , we write eqn.(1) as ,

ΔQ = ΔU + P ΔV ................ (3)

Let us assume a constant volume process and C_vis molar specific heat at constant volume,

Then quantity of heat absorbed by one mole for increase of temperature ΔT is given as

ΔQ = C_v ΔT .......................(4)

ΔV = 0 for constant volume process. Hence we rewrite eqn.(3) using eqn.(4) as

ΔU = ........................ (5)

Above equation (5) is for change in internal energy if the gas temperature is increaed by ΔT .

Let us assume a constant pressure process and C_pis molar specific heat at constant pressure,

Then quantity of heat absorbed by one mole for increase of temperature ΔT is given as

ΔQ = ΔU + P ΔV = C_p ΔT .......................(6)

In above equation , we have expressed the heat absorbed ΔQ in terms of internal energy and

workdone using eqn.(2)

For ideal gas of one mole , we have , P V = R T ................ (7)

By differentiating eqn.(7) , we get , P ΔV + V ΔP = R ΔT ................... (8)

For constant pressure process , ΔP = 0 , hence we get from eqn.(8) ,

P ΔV = R ΔT ................... (9)

Let us use eqn.(5) for internal energy ΔU and use eqn.(9) for workdone and rewrite eqn.(6) as

C_p ΔT = C_v ΔT + R ΔT or C_p = C_v + R .........................(10)

Equation (10) is the relation between specific heat at constant pressure and specific heat at

constant volume of ideal gas .

Answered by Thiyagarajan K | 12 Jan, 2021, 12:22: AM

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