Answer to Question #171847 in Organic Chemistry for Ornyx Oswago

Question

Show that for one mole of a perfect gas, 𝑑𝑆𝑚 = 𝐶𝑃,𝑚

𝑑𝑇

𝑇

− 𝑅

𝑃

𝑑

Answer 1

Entropy is by its definition: dS = (dU+PdV)T

Multiply that by T: TdS - PdV = dU = d(H-PV) = dH - PdV - VdP

So: TdS = dH - VdP

For an ideal gas: PV=nRT or PV_m=RT, dU = c_vdT, dH = c_pdT

Finally, we get dS_m = c_p,mdT/T - RdP/P