Combining the statement written in question 19 with the mathematical statement of the 1st

law of thermodynamics, show in not more than 6 steps that;
du pdv
T
ds  1 

To combine the statement written in question 19 with the mathematical statement of the 1st law of thermodynamics, we need to use some thermodynamic equations and manipulate them to derive the desired expression. Here's how you can do it in not more than 6 steps:

Step 1: Start with the mathematical statement of the 1st law of thermodynamics:
dU = δQ - δW

Step 2: Rearrange the equation to isolate δQ:
δQ = dU + δW

Step 3: Use the definition of enthalpy (H = U + PV) to express the change in internal energy (dU) in terms of the change in enthalpy (dH) and pressure-volume work (PdV):
dU = dH - PdV

Step 4: Substitute the expression for dU in terms of dH and PdV into the equation from step 2:
δQ = dH - PdV + δW

Step 5: Divide both sides of the equation by temperature (T) to obtain the change in entropy (dS) on the left side:
dS = δQ / T = (dH - PdV + δW) / T

Step 6: Simplify the equation to get the final expression:
dS = (dH / T) - (PdV / T) + (δW / T)

This is the desired expression combining the statement from question 19 with the mathematical statement of the 1st law of thermodynamics.