## Example 8.6

$\Delta U=q+w$

Angela 1K
Posts: 80
Joined: Fri Sep 29, 2017 7:05 am

### Example 8.6

Calculate the final temperature and the change in internal energy wen 500. J of energy is transferred as heat to 0.900 mol O2(g) at 298 K and 1.00 atm at (a) constant volume; (b) constant pressure. Treat the gas as ideal.

I understand how the textbook determine CV,m and CP,m, but I don't understand where they get the equations for parts (a) and (b) from. This is the work that the textbook show:

(a) From $\Delta T=q/nCv,m$,
$\Delta T = \frac{500. J}{(0.900 mol) \cdot (20.79 J\cdot K^{-1}\cdot mol^{-1})} = +26.7 K$

Find the final temperature.
T = 298 + 26.7 K = 325 K
From $\Delta U = q$ at constant volume,
$\Delta U = +500. J$

(b) From $\Delta T = q/nCp,m$
$\Delta T = \frac{500. J}{(0.900 mol) \cdot (29.10 J\cdot K^{-1}\cdot mol^{-1}))} = +19.1 K$
Find the final temperature.
T = 298 + 19.1 K = 317 K

Heat at constant volume: Transfer enough energy to raise the temperature to its final value (317 K), and use $\Delta U=q$
$\Delta U = q = (0.900 mol) \cdot (20.79 K\cdot K^{-1}\cdot mol^{-1}) \cdot (19.1 K) = +357J$

Expand at constant temperature: Allow the sample to expand isothermally to its final volume.
Because U is independent of volume for an ideal gas, $\Delta Ustep 2 = 0$

Now add the two changes in internal energy.
$\Delta U = +357J$

Sorry for the long problem but if someone could please explain it to me (step-by-step) regarding why they used each equation and how they got to each step, that would be greatly appreciated. Thank you so much!

Lena Nguyen 2H
Posts: 51
Joined: Fri Sep 29, 2017 7:06 am
Been upvoted: 1 time

### Re: Example 8.6

For part (a), the first formula is the equation $q = n\Delta TC_{V, m}$ from the textbook but modified so that $\Delta T$ is on its own side. From $\Delta T = T_{f} - T_{i}$, the final temperature is calculated.

$\Delta U = q$ at constant volume because no expansion work is done (assuming no nonexpansion work is done either), so w = 0.

Sorry, I'm not sure I completely understand the ideal gas has $\Delta U = 0$ when expanding isothermally in part (b), but I hope this helped with part (a).