TITLE.PM5

GASES AND VAPOUR MIXTURES 417

dharm

\M-therm\Th9-1.pm5

9.5. Specific Heats of a Gas Mixture

— As per Gibbs-Dalton law, the internal energy of a mixture of gases is given by

mu = Σ miui ...(i) [from eqn. (9.5)]

Also u = cvT ...(ii) [from perfect gas equation]

Now from (i) and (ii), we have

mcvT = Σ micviT

∴ mcv = Σ micvi

or cv = m

m

i

∑^ cvi^ ...(9.19)

Similarly from equations, mh = Σ mihi

and h = cpT, we get

mcpT = Σ micpiT

∴ mcp = Σ micpi

or cp = m

m

i

∑^ cpi ...(9.20)

From eqns. (9.18) and (9.19),

cp – cv =

m

m

i

∑^ cpi –

m

m

i

∑^ cvi =

m

m

i

∑ (cpi – cvi)

Also cpi – cvi = Ri, therefore,

cp – cv =

m

m

i

∑^ Ri

Also from eqn. (9.12), R =

m

m

i

∑^ Ri, therefore for the mixture

cp – cv = R

The following equations can be applied to a mixture of gases

γ =

c

c

p

v

; cv =

R

γ− 1

; cp =

γ

γ

R

− 1

It should be borne in mind that γ must be determined from the eqn. γ =

c

c

p

v

; there is no weighted

mean expression as there is for R, cv and cp.
— In problems on mixtures it is often convenient to work in moles and the specific heats can
be expressed in terms of the mole. These are known as molar heats, and are denoted by Cp
and Cv.
Molar heats are defined as follows :
Cp = Mcp and Cv = Mcv ...(9.21)
But cp – cv = R
∴ Cp – Cv = Mcp – Mcv = M(cp – cv) = MR
But MR = R 0
∴ Cp – Cv = R 0 ...(9.22)
Also U = mcvT = mMc T
M

v