Unformatted text preview: 117 4.4. HEAT CAPACITY Consequently, u is not a function of v for an ideal gas, so u = u(T ) alone. Since Eq. (4.78),
h = u + P v , for an ideal gas reduces to h = u + RT
h = u(T ) + RT = h(T ). (4.153) Now return to general equations of state. With s = s(T, v ) or s = s(T, P ), one gets
ds =
ds = ∂s
∂T
∂s
∂T ∂s
dv,
∂v T
∂s
dT +
dP.
∂P T dT +
v P (4.154)
(4.155) Now using Eqs. (4.102, 4.117, 4.141, 4.144) one gets
∂P
cv
dT +
T
∂T
cP
∂v
ds =
dT −
T
∂T dv, ds = (4.156) v dP. (4.157) P Subtracting Eq. (4.157) from Eq. (4.156), one ﬁnds
∂P
cv − cP
∂v
dT +
dv +
T
∂T v
∂T
∂v
∂P
dv + T
dP.
(cP − cv )dT = T
∂T v
∂T P
0= dP, (4.158) P (4.159) Now divide both sides by dT and hold either P or v constant. In either case, one gets
cP − cv = T ∂P
∂T v ∂v
∂T . (4.160) P Also, since ∂P/∂T v = −(∂P/∂v T )(∂v/∂T P ), Eq. (4.160) can be rewritten as
cP − cv = −T ∂v
∂T 2
P ∂P
∂v . (4.161) T Now since T > 0, (∂v/∂T P )2 > 0, and for all known materials ∂P/∂v T < 0, we must have
c P > cv . (4.162) Example 4.7
For an ideal gas ﬁnd cP − cv .
CC BYNCND. 18 November 2011, J. M. Powers. 118 CHAPTER 4. MATHEMATICAL FOUNDATIONS OF THERMODYNAMICS
For the ideal gas, P v = RT , one has
∂P
∂T =
v ∂v
∂T R
,
v =
P R
.
P (4.163) So, from Eq. (4.160), we have
cP − cv =
=
= RR
,
vP
R2
,
T
RT
R. T (4.164)
(4.165)
(4.166) This holds even if the ideal gas is calorically imperfect. That is
cP (T ) − cv (T ) = R. (4.167) For the ratio of speciﬁc heats for a general material, one can use Eqs. (4.141) and (4.144)
to get
cP
k=
cv =
=
=
= T ∂s
∂T P
∂s
∂T v T
∂s
∂T , then apply Eq. (4.56) to get ∂T
, then apply Eq. (4.58) to get
P ∂s v
∂s
∂T ∂v
∂P
−
−
,
∂P T ∂T s
∂v s ∂s T
∂v
∂ P ∂T
∂s
.
∂s T ∂P T
∂T s ∂v s (4.168)
(4.169)
(4.170)
(4.171) So for general materials
k= ∂v
∂P T ∂P
∂v . (4.172) s The ﬁrst term can be obtained from P − v − T data. The second term is related to the
isentropic sound speed of the material, which is also a measurable quantity.
Example 4.8
For a calorically perfect ideal gas with gas constant R and speciﬁc heat at constant volume cv ,
ﬁnd expressions for the thermodynamic variable s and thermodynamic potentials u, h, a, and g , as
functions of T and P .
CC BYNCND. 18 November 2011, J. M. Powers. ...
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 Fall '11
 ParkSou
 Dynamics, Thermodynamics, ∂t, J. M. Powers, cp − cv

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