download the script: Entropy change of ideal gases

Recall that entropy, which is defined as δq_rev/T, is a property of state. So if we want to calculate the entropy change ΔS, we can arbitrarily choose some imaginary reversible processes from initial state to final state.

We write down here the definition of entropy:

T·ds=δq_rev

According to the 1.Law of thermodynamics for reversible process which is expressed as:

du=δq_rev-p·dν

we can rewrite the equation of entropy by using the 1.Law of thermodynamics:

T·ds= du+p·dν

Also, the definition of enthalpy is:

h=u+p·ν

→dh=d(u+p·ν)=du+ p·dν+ ν·dp

Therefore another equation of T·ds is expressed by using enthalpy:

T·ds= dh-ν·dp

Now we obtain two important equations from above derived expressions, namely:

T·ds= du+p·dν

T·ds= dh-ν·dp

Both equations are known as Gibbs Equation.

If gas is assumed to be ideal gas, we have additional the ideal gas law:

p·ν=R·T

Hence, the expressions of entropy change of an ideal gas can be calculated from both Gibbs equations and ideal gas law:

• From T·ds= du+p·dν, we have:

• From T·ds= dh-ν·dp, we have:

• If pressure p and volume per unit mass ν are given:

So if any two among these three properties, pressure p, temperature T and volume per unit mass ν (or volume V), are given, we can choose one of above three equations to determine the entropy change from state 1 to state 2, regardless whether the process reversible or irreversible is: