Ch.%204%20

1
Ch. 4 1st Law of Thermodynamics

• Heat Capacities
• Consider a homogeneous system of constant
  composition
• Write dU (dH) as a total differential of the
  independent variables X and Y (for our purposes,
  X and Y could be any pair of p, V, and T). Then
• From the 1st Law, ?Q dU pdV, we can
  substitute to get
• where X and Y are T and V, respectively.

2
Ch. 4 1st Law of Thermodynamics

• Heat Capacities
• From ?Q dH Vdp we can substitute for dH to
  get
• where X and Y are T and p, respectively.
• For a process of heating at V const we get
• or per unit mass we can write

3
Ch. 4 1st Law of Thermodynamics

• Heat Capacities
• For a process of heating at p const we get
• or per unit mass we can write
• Can also calculate heat of change of V and p at T
  const
• Not very useful, but given for completeness

4
Ch. 4 1st Law of Thermodynamics

• Calculation of Internal Energy and Enthalpy
• Our equations for CV and Cp can be integrated
  directly for processes with V const and p
  const, respectively to find U and H, if CV and Cp
  are known as functions of T
• CV and Cp, from experiment, are usually
  polynomials in T

5
Ch. 4 1st Law of Thermodynamics

• Calculation of Internal Energy and Enthalpy
• Consider 2 rigid vessels linked by a connection
  with a stopcock (pg. 34). One contains gas, the
  other evacuated.
• Stopcock opened, gas in 1 expands to occupy both
  vessels.
• Temperature measurements show that system
  exchanges no heat with environment ? no work
  done, Q 0, W 0, and ?U 0.
• Since p changed during process, we have U U(T)
  only and partial derivative used above are total
  derivatives.

6
Ch. 4 1st Law of Thermodynamics

• Calculation of Internal Energy and Enthalpy
• Notes on this experiment
• When experiment done carefully, small heat
  exchange was found (Joule-Thomson effect), which
  vanishes for ideal gas behavior
• As gas confined in vessel 1 expands into 2, work
  is done by some portions of gas against others
  while volumes change (as molecules enter 2, they
  are effected by molecules following)
• These are internal transfers that are not
  included in W
• This shows the importance of defining system
  carefully and clearly when considering a
  thermodynamic process
• In this case, the system is all the gas contained
  in both vessels (initially one is empty), whose
  total volume (V1 V2) does not change

7
Ch. 4 1st Law of Thermodynamics

• More on Heat Capacities
• As noted above, since U U(T) we have CV dU/dT
  and cv du/dT
• We can write H U pV U nRT H(T)
  leading to Cp dH/dT and cp dh/dT
• Since we are only interested in differences in
  internal energy and enthalpy, we can set the
  integration constant to 0 giving U ? CVT, H ?
  CpT, u ? cvT, and h ? cpT

8
Ch. 4 1st Law of Thermodynamics

• More on Heat Capacities
• Since we have CV dU/dT, Cp dH/dT and H U
  pV U nRT, we have
• leading to Cp CV nR ? cp cv R recalling
  that n m/M.
• As mentioned earlier, heat capacities for all
  gases can be measured and the coefficients for
  the polynomial expansion can be determined (C ?
  ?T ?T2 )

9
Ch. 4 1st Law of Thermodynamics

• More on Heat Capacities
• For simple gases like N2, O2, and Ar, the
  experimental data are nearly constant for all
  temperatures and pressures of interest, so the
  temperature variation is not considered.
• From earlier we have, for monatomic gases, the
  total internal energy is U (3/2)NkT, which
  leads to CV (3/2)nR and cv (3/2)R.
  Similarly, Cp (5/2)nR and cp (5/2)R.
• For diatomic gases, where there are more degrees
  of freedom, so we get CV (5/2)nR and cv
  (5/2)R. Similarly, Cp (7/2)nR and cp
  (7/2)R. The ratios Cp/CV cp/cv ?.

10
Ch. 4 1st Law of Thermodynamics

• More on Heat Capacities
• Dry air is considered to be a diatomic gas, so
  the second form applies.
• The ratio, cp/cv ? 1.4.
• We then attach the subscript, d, to the specific
  heats to indicate dry air.
• This leads to cvd 718 J kg-1 K-1 and cpd 1005
  J kg-1 K-1 and Rd cpd cvd 287(.05) J kg-1
  K-1.

11
Ch. 4 1st Law of Thermodynamics

• More on Heat Capacities
• The table below shows the values of cpd for
  various temperatures and pressures. Note the
  slight variation.

12
Ch. 4 1st Law of Thermodynamics

• More Forms of the 1st Law
• Using the above expressions for (specific) heat
  capacities, we get more useful forms of the 1st
  Law, two of which are particularly useful
• and

13
Ch. 4 1st Law of Thermodynamics

• Special Cases
• For an isothermal transformation
• For an isochoric (constant volume) transformation
• For an isobaric transformation