How to Derive PdV=RdT/(1-δ) from the Polytropic Process Constant Equation?

[HethensEnd25]

Homework Statement

One mole of an ideal gas is reversibly compressed from 1 bar to P2
bar that results in temperature increase from 400 K to 950 K. Calculate the heat
transferred during the process and the final pressure, if the path followed by the gas can
be given by PV^1.55 = , and the molar heat capacity of the gas is given by

Cp/R = 3.85 + 0.57×10^-3T (where T is in K)

Homework Equations

dU=dQ+dW

The Attempt at a Solution

my attempt at a solution.

1. I used the first law of thermodynamics to get

dU=dQ+dW

solving for dQ

dQ=dU-dW ===> dQ=CvdT+PdV

2. Using Ideal gas solved for T then differentiated

RdT=PdV +VdP

3. It was given that PV^1.55=Constant

I know since it is a real number I can use steps of a polytropic process

PV^δ=Constant

This is where I am getting hung up. My books steps to solution say that using


RdT=PdV +VdP solving for PdV

is the next step to solve the above expression PV^δ=Constant.

It then says that PδV^δ-1dV=-V^δdP from which VdP=-PδdV

My question is how did they derive that expression from the PV^δ=Constant

the end result being that

PdV=RdT/(1-δ)


Any and all help clarifying this for me would be greatly appreciated I have other formula that are all plug and chug but I would rather understand the steps involved in solving the problem with these constants.

Best Regards,

D

 

[vela]

This is where I am getting hung up. My books steps to solution say that using

RdT=PdV +VdP solving for PdV

is the next step to solve the above expression PV^δ=Constant.

What do you mean by "solve the above expression"? Solve it for what?

It then says that PδV^δ-1dV=-V^δdP from which VdP=-PδdV

My question is how did they derive that expression from the PV^δ=Constant

Hint: Consider the multivariable function $f(p,V) = \text{constant}$. If you differentiate it, you get
$$df = \frac{\partial f}{\partial p}dP + \frac{\partial f}{\partial V}dV = 0.$$

 

[HethensEnd25]

What do you mean by "solve the above expression"? Solve it for what?

I am trying to use that PV^δ=constant

The book uses that to get the term of work into a term where the n factor can be used.

Here is a picture of part of the solution I am referring to and it has the part in question highlighted

My question is how are they coming to that answer for work from that expression? Is it from your hint?

 

[HethensEnd25]

Hint: Consider the multivariable function f(p,V)=constantf(p,V) = \text{constant}. If you differentiate it, you get

df=∂f∂pdP+∂f∂VdV=0.​

This is what I was looking for thank you!