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Thermodynamics filled in class notes_Part_55

# Thermodynamics filled in class notes_Part_55 - 117 4.4 HEAT...

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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 BY-NC-ND. 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 BY-NC-ND. 18 November 2011, J. M. Powers. ...
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