Equation (38.24) does not tell us about the direction (i.e., a hot body with respect to its surrounding will gain temperature or cool down) of the process. To determine the proper direction of a process, we define a new state variable, entropy, which is
(38.25)
where s is the entropy of the system,
is the heat added reversibly to the system and
T is the temperature of the system. It may be mentioned that Eqn. (38.25) is valid if both external and internal irreversibilities are maintained during the process of heat addition
Entropy is a state variable and it can be associated with any type of process, reversible or irreversible. An effective value of
can always be assigned to relate initial and end points of an irreversible process, where the actual amount of heat added is
. One can write
(38.26)
It states that the change in entropy during a process is equal to actual heat added divided by the temperature plus a contribution from the irreversible dissipative phenomena. It may be mentioned that dsirrev implies internal irreversibilities if T is the temperature at the system boundary. If T is the temperature of the surrounding dsirrev implies both external and internal irreversibilities. The irreversible phenomena always increases the entropy, hence
(38.27)
Significance of greater than sign is understandable. The equal sign represents a reversible process. On combining Eqs (38.26) and (38.27) we get.
(38.28)
If the process is adiabatic,
, Eq. (38.28) yields
(38.29)
Equations (38.28) and (38.29) are the expressions for the second law of thermodynamics. The second law tells us in what direction the process will take place. The direction of a process is such that the change in entropy of the system plus surrounding is always positive or zero (for a reversible adiabatic process). In conclusion, it can be said that the second law governs the direction of a natural process.
For a reversible process, it can be said that
where
is change in volume and from the first law of thermodynamics it can be written as
(38.30)
If the process is reversible, we use the definition of entropy in the form
the Eq. (38.30) then becomes,
(38.31)
Another form can be obtained in terms of enthalpy. For example, by definition
Differentiating, we obtain
(38.32)
Combining Eqs (38.31) and (38.32), we have
(38.33)
Equations (38.31) and (38.33) are termed as first Tds equation and
second Tds equation, respectively.
For a thermally perfect gas, we have
(from Eq. 38.20) , substitute this in Eq. (38.33) to obtain
(38.34)
Further substitution of
into Eq. (38.34) yields
(38.35)
Integrating Eq. (38.35) between states 1 and 2,
(38.36)
If
is a variable, we shall require gas tables; but for constant
, we obtain the analytic expression
(38.37)
In a similar way, starting with Eq. (38.31) and making use of the relation
the change in entropy can also be written as
is the heat added reversibly to the system and
. One can write
, Eq. (38.28) yields
where
is change in volume and from the first law of thermodynamics it can be written as
the Eq. (38.30) then becomes,
(from Eq. 38.20) , substitute this in Eq. (38.33) to obtain
into Eq. (38.34) yields 
is a variable, we shall require gas tables; but for constant
the change in entropy can also be written as 
