THERMODYNAMICS  THEORY



Internal Energy, Enthalpy, and Specific Heats of Ideal Gas

Ideal Gas Model 
Pv = PT 
u = u(T) 
h = h(T) = u(T) + RT 
Ideal Gas Model


The ideal gas is defined as a gas which obeys the following equation of state:
Pv = RT
The internal energy of an ideal gas is a function of temperature only. That is,
u = u(T)
Using the definition of enthalpy and the equation of state of ideal
gas to yield,
h = u + P v = u + RT
Since R is a constant and u = u(T), it follows that the enthalpy of
an ideal gas is also a function of temperature only.
h = h(T)
Since u and h depend only on the temperature for an ideal gas, the constant
volume and constant pressure specific heats c_{v} and c_{p} also
depend on the temperature only.
c_{v} = c_{v} (T) c_{P} = c_{P} (T)
For an ideal gas, the definitions of c_{v} and c_{p} are
given as follows:
and they can be rewritten as
c_{v} = du/dT c_{P} = dh/dT
During a process from state 1 to state 2, the changes of internal energy
and enthalpy are:




Internal Energy 
Enthalpy 


Three Ways of Calculating Δu and
Δh
Specific Heats of
Common Ideal Gases


There are three ways to determine the changes of internal energy and enthalpy for ideal gas.
 By using the tabulated u and h data. This is the easiest and most accurate way.
 By integrating the equations above if the relations of c_{v }and
c_{p} as a function of temperature are known. This is for computerized
calculations and very accurate.
 By using the average values of specific heats. This is simple and
the results obtained are reasonably accurate if the temperature interval is small.






Specific Heat Relations of Ideal Gas



Combing the definition of enthalpy and the equation
of state for ideal gas to yield
h = u + pv = u + RT
Differentiating the above relation to give
dh = du + RdT
Replacing dh by c_{P}dT and du by c_{v}dT,
This is a special relationship between c_{v} and c_{P }for
an ideal gas. Also, the ratio of c_{P} and c_{v} is
called the specific heat ratio,
k = c_{P}/c_{v}
The specific heat ratio is also a temperature dependent property.
For air at T = 300 K,
c_{P }=
1.005 kJ/(kgK)
c_{v} = 0.718 kJ/(kgK)
k =
1.4






The Polytropic Process_{}

Special Processes of Ideal Gas on a Pv Diagram


Many processes which occur in practice can be described by an equation
of the form
Pv^{n }= constant
where
n = constant
For a process from state
1 to state 2, the relation is:
P_{1}v_{1 }^{n} =
P_{2}v_{2}^{n }
By
using the equation of state of ideal gas, the relations between P, v,
and T are:
T_{2}/T_{1}=(P_{2}/P_{1})^{(n1)/n } P_{2}/P_{1}=(v_{1}/v_{2})^{n}
For some specific values of n, the process becomes isobaric, isothermal,
isometric, and adiabatic, and they are summarized as follows:
Process 
n 
Isobaric (P = constant) 
0 
Isothermal (T = constant) 
1 
Isometric (V = constant) 
infinity 
adiabatic (Q = 0) 
k 






Energy Analysis for Ideal Gas in a Closed System at
Different Processes_{}

Boundary Work


The energy balance for a stationary, closed system is:
Q  W = ΔU
For ideal gas, the internal energy can be determined by
ΔU = U_{2}  U_{1} = c_{v}(T_{2}  T_{1})
where
c_{v} = the average value of constant volume
specific heat
If only boundary work is considered, then the work W can be determined by
For a polytropic process, the work W is:
For special processes such as the isobaric, isothermal,
isometric, and adiabatic processes for ideal gas, using the average specific
heats, the heat, work, and internal energy are given in the following table:




Process 
Q 
W 
ΔU 
Isothermal 
mRT ln(P_{1}/P_{2}) 
mRT ln(P_{1}/P_{2}) 
0 
Isometric 
m c_{v} ΔT 
0 
mc_{v}ΔT 
Isobaric 
mc_{P} ΔT 
P(V_{2}V_{1}) 
mc_{v}ΔT 
Adiabatic 
0 
mc_{v}ΔT 
mc_{v}ΔT 



