Temperature change in a gas expansion

[yamata1]

Homework Statement

The fluid is a perfect gas. Constant pressure heating is broken down into two stages:
- an elementary transformation AB which is a heating at constant volume, during which
temperature and pressure vary by $\delta T_V$ and $\delta P_V$.

-an elementary transformation BC which is a reversible adiabatic relaxation, during
which the temperature and the pressure vary respectively from $\delta T_S$ and $\delta P=-\delta P_V$

1-Express $\delta P_V$ as a function of $\delta T_V$,T and P

2-Express $\delta T_S$ as a function of $\delta P_V$,T , P and $\gamma=\frac{C_p}{C_v}$ then as a function of $\delta T_V$ and $\gamma$.

Relevant Equations

$PV^{\gamma}=cst\; \; \; \; \;TV^{\gamma -1}=cst$ and $TP^{\frac{1-\gamma }{\gamma}}=cst$

PV=nRT , dU=TdS-PdV ,$\delta Q=C_V\deltaT$ , $\delta V =(\frac{\partial V}{\partial T})_P \delta T+(\frac{\partial V}{\partial P})_T \delta P$

1- $\delta P_V =(\frac{\partial P}{\partial T} )_V \delta T_V$

2-$\delta V =(\frac{\partial V}{\partial T})_P \delta T+(\frac{\partial V}{\partial P})_T \delta P$ so $C_v \delta T=-P\delta V=-P((\frac{\partial V}{\partial T})_P \delta T+(\frac{\partial V}{\partial P})_T \delta P)$ I can replace $-P((\frac{\partial V}{\partial T})_P \delta T)=nR\delta T$ since we have an ideal gas and make $\gamma$ appear that way.

Is there some other equation I am forgetting ?

Thank you

 

[Chestermiller]

Using the ideal gas law, what is your actual answer for part 1 (not in terms of partial derivatives)?

For part 2, you don't need to use the first law if you use $TP^{\frac{1-\gamma }{\gamma}}=cst$.

 

[yamata1]

Using the ideal gas law, what is your actual answer for part 1 (not in terms of partial derivatives)?

For part 2, you don't need to use the first law if you use $TP^{\frac{1-\gamma }{\gamma}}=cst$.

$\delta P_V =(\frac{\partial P}{\partial T} )_V \delta T_V =\frac{nR}{V}\delta T_V$.
I don't know how to use $TP^{\frac{1-\gamma }{\gamma}}=cst$ to make $\delta T_S$ a function of $\delta P_V$,T,P and $\gamma$.

 

[Chestermiller]

$\delta P_V =(\frac{\partial P}{\partial T} )_V \delta T_V =\frac{nR}{V}\delta T_V$.

I get $$\delta T_V=\frac{\delta P}{P}T$$

I don't know how to use $TP^{\frac{1-\gamma }{\gamma}}=cst$ to make $\delta T_S$ a function of $\delta P_V$,T,P and $\gamma$.

I would write algebraically, $$(T+\delta T_V+\delta T_S)P^{\frac{1-\gamma }{\gamma}}=(T+\delta T_V)(P+\delta P)^{\frac{1-\gamma }{\gamma}}$$

 

[yamata1]

I would write algebraically, $$(T+\delta T_V+\delta T_S)P^{\frac{1-\gamma }{\gamma}}=(T+\delta T_V)(P+\delta P)^{\frac{1-\gamma }{\gamma}}$$

Thank you but now I don't see a way to remove T and have $\delta T_S$ as a function of only $\delta T_V$ and $\gamma$.

 

[Chestermiller]

Thank you but now I don't see a way to remove T and have $\delta T_S$ as a function of only $\delta T_V$ and $\gamma$.

What does "Express δ$P_V$ as a function of δ$T_V$,T and P" and "Express $\delta T_S$ as a function of $\delta P_V$,T , P and $\gamma$" mean to you?

 

[yamata1]

What does "Express δ$P_V$ as a function of δ$T_V$,T and P" and "Express $\delta T_S$ as a function of $\delta P_V$,T , P and $\gamma$" mean to you?

Since δ$T_V$ is a function of T and P there should be a way to change variables P and T in the formula for $\delta T_S$ and express it as a function of only δ$T_V$ and $\gamma$.

 

[Chestermiller]

Since δ$T_V$ is a function of T and P there should be a way to change variables P and T in the formula for $\delta T_S$ and express it as a function of only δ$T_V$ and $\gamma$.

Who says?

 

[yamata1]

Who says?

Question 2 :
Express $\delta T_S$ as a function of $\delta P_V$,T , P and $\gamma$ first then as a function of $\delta T_v$ and $\gamma$

 

[Chestermiller]

Question 2 :
Express $\delta T_S$ as a function of $\delta P_V$,T , P and $\gamma$ first then as a function of $\delta T_v$ and $\gamma$

The "then" part can't be done unless T is also included. This is just an omission from the problem statement.

 

[yamata1]

The "then" part can't be done unless T is also included. This is just an omission from the problem statement.

I guess so.Thank you for these clarifications.