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

In summary, the conversation discusses a problem in thermodynamics involving an insulated cylinder with a frictionless piston and an attached spring. The cylinder is connected to a mains pipe and air flows in at high pressure and temperature. The first law of thermodynamics is applied to calculate the final pressure, temperature, and mass of the air in the cylinder. The conversation also covers the use of equations and assumptions in solving the problem.
  • #36
Tomdickjerry said:
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 Cp 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 CpdT.

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
 
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  • #37
Chestermiller said:
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 Cp 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 CpdT.

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?
 
  • #38
Tomdickjerry said:
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 Cp expressed exclusively in terms of R?
 

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