Slope of constant volume and constant pressure processes in T-s plot

# For constant pressure processes(p=C),

ds = m . Cp . dT/T ---------------------- 1.

T-s Plot

# The slope of the curve in T-s plot is dT/dS

Therefore from equation 1.,  dT/dS = T/(m . Cp) ---------------- 2.

For 1 kg of perfect gas,  

Equation 2. becomes

dT/dS = T/Cp    {Since m = 1 kg} -------------------- 3.


# Similarly for constant volume processes(v=C),  

dT/dS = T/Cv --------------------- 4.

# For perfect gas, we know that

Cp > Cv 

or

1/Cp < 1/Cv  --------------------- 5.

# Compare equation 5. with equation 2. & 4.

T/Cp  <  T/Cv

or

{dT/dS} of v=C > {dT/dS} of p=C ----------------6.

Here {dT/dS} is slope of the curve in T-s plot.

# Therefore the slope of the curve for constant volume process (a-b) is higher than that of constant pressure processes (a-b’). 




Note ;

Constant volume process , v=C.

Constant pressure process , p=C.

Cp, Cv - Specific heats at constant pressure and constant volume.

ds - Change in entropy.