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348 ENGINEERING THERMODYNAMICS

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Fig. 7.2. Determination of compressibility from p-T data.

K = –^1

v

v

pT

∂

∂

F

HG

I

KJ

...(7.36)

K can be regarded as a constant for many purposes for solids and liquids. In tables of

properties it is often quoted as an average a value over a small range of pressure at atmospheric

temperature, i.e.,

K = – vv

vp p

21

12 1

−

()−

When β and K are known, we have

∂

∂

F

HG

I

KJ

∂

∂

F

HG

I

KJ

∂

∂

F

HG

I

KJ

p

T

T

v

v

v p pT

= – 1

Since

∂

∂

F

HG

I

KJ

v

T p

= βv and ∂

∂

F

HG

I

KJ

v

pT

= – Kv,

∂

∂

F

HG

I

KJ

p

T v

= β

K

...(7.37)

When the equation of state is known, the co-efficient of cubical expansion and compressibility

can be found by differentiation. For a perfect gas, for example, we have

∂

∂

F

HG

I

KJ

v

T p =

R

p

v

p

RT

T p

and

∂

∂

F

HG

I

KJ

= 2

Hence β =

1

v

v

T p

∂

∂

F

HG

I

KJ =

R

pv

=^1

T

,

and K = –^1

v

v

pT

∂

∂

F

HG

I

KJ

=

RT

pv^2

=

1

p

7.6.3. Specific heats

Following are the three differential co-efficients which can be relatively easily determined

experimentally.