Deriving First Law of Thermo Equations

[mortalapeman]

Homework Statement

This is really just a question that I can't seem to find a good solution for in my book. Basically I'm trying to understand for the first law of thermodynamics how you can derive the equation in term of P1 and P2. I don't understand how to go from PdV to (something)dP. This assuming we are dealing with an ideal gas.

Homework Equations

$$PV = RT$$

$$C_{v} = C_{p} - R$$

$$\Delta U = Q + W$$

$$W = -PdV$$

$$dQ = C_{v}dT + RTV^{-1}dV$$

$$W = -PdV = -RTV^{-1}dV = -RT \ln \left ( V_{1}/V_{2} \right ) = RT \ln \left ( P_{2}/ P_{1} \right )$$

The Attempt at a Solution

I understand how to get to:

$$dQ = C_{v}dT + RTV^{-1}dV$$

and there is another equation in my book that is:

$$dQ = C_{p}dT - RTP^{-1}dP$$

with work equal to:

$$dW = -RdT + RTP^{-1}dP$$

And its getting to those equations that i don't understand how to do. Any help in the right direction would be appreciated :)

 

[Andrew Mason]

I understand how to get to:

$$dQ = C_{v}dT + RTV^{-1}dV$$

and there is another equation in my book that is:

$$dQ = C_{p}dT - RTP^{-1}dP$$

with work equal to:

$$dW = -RdT + RTP^{-1}dP$$

And its getting to those equations that i don't understand how to do. Any help in the right direction would be appreciated :)

Start with:

PV = RT (we will assume n=1)

Differentiate with respect to T:

d(PV)/dT = R = P(dV/dT) + V(dP/dT)

So: RdT = PdV + VdP = dW + VdP

Which means that: dW = RdT - VdP = RdT - RTdP/P

AM

 

[mortalapeman]

Start with:

PV = RT (we will assume n=1)

Differentiate with respect to T:

d(PV)/dT = R = P(dV/dT) + V(dP/dT)

So: RdT = PdV + VdP = dW + VdP

Which means that: dW = RdT - VdP = RdT - RTdP/P

AM

Thanks so much for clearing that up for me. These things always seem to be so simple, can't believe I didn't see that xD

 

[Andrew Mason]

Thanks so much for clearing that up for me. These things always seem to be so simple, can't believe I didn't see that xD

Note: I am using dW as the work done BY the gas. I see you are using dW is the work done ON the gas, so dW = -PdV. Most texts now use dW = PdV. It is much less confusing.

AM