What is the equation for Cp in terms of R for an adiabatic charging process?

[Tomdickjerry]

As you responded in a private conversation: W=[kx2/2+PaAx]. So now what do we get if we substitute this into our equation:
$$m(h-h_{in})=-W+kx^2+P_aAx$$
Chet

m(h−hin)=−[kx²/2+PaAx]+kx²+PaAx ?

 

[Chestermiller]

m(h−hin)=−[kx²/2+PaAx]+kx²+PaAx ?

Collect terms?

 

[Tomdickjerry]

Collect terms?

m(h−hin)=1/2kx²?

 

[Chestermiller]

Correct. Pretty simple, huh?

This is all that needs to be done on the right hand side of the equation. I'll be back in a little while to get you started working on the left hand side of the equation. Right now, I'm on my iPhone.

 

[Tomdickjerry]

Correct. Pretty simple, huh?

This is all that needs to be done on the right hand side of the equation. I'll be back in a little while to get you started working on the left hand side of the equation. Right now, I'm on my iPhone.

alright thanks! for the left side do i just use cv(t₂-t₁) i figured since the it isn't constant pressure process

 

[Chestermiller]

alright thanks! for the left side do i just use cv(t₂-t₁) i figured since the it isn't constant pressure process

Actually, no, but you're on the right track. The molar enthalpy of a pure single phase substance can be expressed uniquely as a function of temperature T and pressure P: h = h(T,P). The molar heat capacity C_p is, by definition, the partial derivative of h with respect to T at constant P:
$$C_p\equiv \left(\frac{\partial h}{\partial T}\right)_P$$For an ideal gas, molar enthalpy is a function only of temperature T. So the partial derivative becomes a total derivative: $$\frac{dh}{dT }=C_p$$For an ideal gas, this equation applies even if the pressure is not constant. The quantity $C_p$ is called the heat capacity at constant pressure only because, in measuring $C_p$ experimentally, not only is the change in enthalpy dh at constant pressure equal to $C_pdT$, but so also is the amount of heat added dQ. So, under constant pressure conditions, $C_p$ can be measured experimentally by directly determining the amount of heat added. If the pressure is not constant, then dQ is not equal to $C_pdT$ but dh (for an ideal gas) is still equal to C_pdT.

So, since for an ideal gas, dh is always equal to $C_pdT$, irrespective of whether the pressure is constant, we have:
$$(h-h_{in})=C_p(T-T_{in})$$where h is the final molar enthalpy of the air in the cylinder, $h_{in}$ is the molar enthalpy of the air in the tank, T is the final temperature of the air in the cylinder, and $T_{in}$ is the temperature of the air in the tank (27 C = 300 K).

Now m is the final number of moles of air in the cylinder. From the ideal gas law, please (algebraically) express m as a function of PV, R, and T, where R is the universal gas constant.

Air is a diatomic gas. In the ideal gas region, how is the molar heat capacity $C_p$ of a diatomic ideal gas related to the ideal gas constant R?

Chet

 

[Tomdickjerry]

Actually, no, but you're on the right track. The molar enthalpy of a pure single phase substance can be expressed uniquely as a function of temperature T and pressure P: h = h(T,P). The molar heat capacity C_p is, by definition, the partial derivative of h with respect to T at constant P:
$$C_p\equiv \left(\frac{\partial h}{\partial T}\right)_P$$For an ideal gas, molar enthalpy is a function only of temperature T. So the partial derivative becomes a total derivative: $$\frac{dh}{dT }=C_p$$For an ideal gas, this equation applies even if the pressure is not constant. The quantity $C_p$ is called the heat capacity at constant pressure only because, in measuring $C_p$ experimentally, not only is the change in enthalpy dh at constant pressure equal to $C_pdT$, but so also is the amount of heat added dQ. So, under constant pressure conditions, $C_p$ can be measured experimentally by directly determining the amount of heat added. If the pressure is not constant, then dQ is not equal to $C_pdT$ but dh (for an ideal gas) is still equal to C_pdT.

So, since for an ideal gas, dh is always equal to $C_pdT$, irrespective of whether the pressure is constant, we have:
$$(h-h_{in})=C_p(T-T_{in})$$where h is the final molar enthalpy of the air in the cylinder, $h_{in}$ is the molar enthalpy of the air in the tank, T is the final temperature of the air in the cylinder, and $T_{in}$ is the temperature of the air in the tank (27 C = 300 K).

Now m is the final number of moles of air in the cylinder. From the ideal gas law, please (algebraically) express m as a function of PV, R, and T, where R is the universal gas constant.

Air is a diatomic gas. In the ideal gas region, how is the molar heat capacity $C_p$ of a diatomic ideal gas related to the ideal gas constant R?

Chet

m=pv/rt.
i don't know what a diatomic gas is it wasn't covered in my syllabus but I am assuming you're referring to the k=1.4?

 

[Chestermiller]

m=pv/rt.
i don't know what a diatomic gas is it wasn't covered in my syllabus but I am assuming you're referring to the k=1.4?

Yes. So, if $\frac{C_p}{C_v}=1.4$ and $C_p-C_v=R$ (you are familiar with this equation, correct?), what is the equation for C_p expressed exclusively in terms of R?