Molar Heat Capacities and Specific Heats for Ideal Gases

[Nachore]

Homework Statement

a. Consider an ideal gas being heated at constant volume, and let Cv be the gas's molar heat capacity at constant volume. If the gas's infinitesimal change in temperature is dT, find the infinitesimal change in internal energy dU of n moles of gas.
Express the infinitesimal change in internal energy in terms of given quantities.

b. Now suppose the ideal gas is being heated while held at constant pressure p. The infinitesimal change in the gas's volume is dV, while its change in temperature is dT. Find the gas's molar heat capacity at constant pressure, Cp.
Express in terms of some or all of the quantities Cv, p, dV, n, and dT.

c. Suppose there are n moles of the ideal gas. Simplify your equation for Cp using the ideal gas equation of state: pV = nRT.
Express Cp in terms of some or all of the quantities Cv, n, and the gas constant R.

d. The ratio of the specific heats Cp/Cv is usually denoted by the Greek letter $$\gamma$$. For an ideal gas, find $$\gamma$$.
Give your answer in terms of some or all of the quantities n, R, and Cv.

Homework Equations

I don't know.

The Attempt at a Solution

For part a, I did dU = nCvdT, but I don't know if it's right.
I'm having trouble approaching rest of the parts. Help please?

 

[Mapes]

Part (a) looks fine. When you increase the internal energy of an ideal gas, its temperature increases, and the constant of proportionality is the heat capacity.

At constant volume, all the energy you added went to increase the internal energy of the gas. But at constant pressure, the gas is allowed to expand, which means it's going to do some $p\,dV$ work on the environment. Now the energy you put in will be divided between this work and increasing the internal energy ($dU$). You'll have both $dU$ and $p\,dV$ where you used to have just $dU$. Does this help?

 

[Nachore]

Umm kind of. But I'm having the most trouble from part b through part d.

 

[Mapes]

My second paragraph is about part (b). Where are you stuck there?

 

[Nachore]

Thanks for part b. But I don't get part c, d.

 

[Mapes]

OK, have you applied the ideal gas law to part (b)?

 

[Nachore]

yes, i got it. thanks

 

[Nachore]

how do you do part d? does anyone know?

 

[Nachore]

Help?

 

[Mapes]

What do you have so far?

 

[Nachore]

I know $$\gamma$$ = Cp/Cv
and
Cv = R/($$\gamma$$ - 1)

How do I find n?

 

[Mapes]

$\gamma$ is independent of n.