From equation (9.3) we have

ds = du / t + pu /t

= cu dt / t + R du / u

Entropy change between state 1-2

or s_{2} – s_{1} = c_{2} In T_{2} / T_{1} + R In v_{2} /v1

we also have Tds = dh udp

ds = dh / T– udp /T

= c_{p} dt / t – R dp /p

Entropy change between two states 1 – 2 can also be written as

s_{2} – s_{1} = cp In T_{2} / T_{1} –R In P_{2} / P_{1}

As c_{p} – c_{u} = R, equation (9 .19) can be written as

s_{2} – s_{1} = cp In T_{2} / T_{1 }– cp In P_{2} P_{1} + c_{2} In P_{2} / P_{1}

or s_{2} –s_{1} = cp In P_{1} T_{1} / P_{2} T_{2} + c_{2} In P_{2} / P_{1}

or s_{2} – s_{1} = cp In v_{2} / v_{1} + c_{2} In p_{2} / p_{1}

or s_{2} – s_{1} = cp In v_{2} / v_{1} + cv In p_{2} / p_{1}

_{ }

We can use equations (9.18), (9.19), (9.20) for computing the e::1tropy change between

two states of an ideal gas.