## Why internal energy is constant when temperature is constant

$\Delta U=q+w$

devenpatel2O
Posts: 26
Joined: Fri Sep 20, 2013 3:00 am

### Why internal energy is constant when temperature is constant

Hello all, I am looking for some help understanding why a system at constant temperature has no change in internal energy. Say a reaction is happening in a container submerged in a heat bath that keeps the reaction at 25 degrees centigrade. Well, change in internal energy is equal to q + w. This reaction will have a q and it will have a w (it's not in a bomb calorimeter), but according to lecture today q and w will be equal and opposite. I've been looking for the answer to why this is true, either conceptually or mathematically (ideally both.) If anyone has any insight, please reply.

Thanks,

Deven

Sungho Son 1G
Posts: 11
Joined: Fri Sep 26, 2014 2:02 pm

### Re: Why internal energy is constant when temperature is cons

Although I don't know how to mathematically show how q = -w, I found some conceptual explanations why ΔU is 0 in an isothermal process (for ideal gases).

The textbook (pages 237, 248) explains how internal energy is the total store of energy in a system. Therefore, ΔU would be the sum of the changes in kinetic and potential energies of the molecules in a system; however, the gas is assumed to be ideal, so potential energy does not exist. Since the temperature is constant, the kinetic energy also remains constant (since temperature and kinetic energy are directly proportional), meaning that the change in kinetic energy is also 0. As a result, ΔU = 0.

Also, ΔU = 3/2 nRΔT, so when temperature is constant, ΔU = 3/2 nR(0) and ΔU = 0.

Since it was found out that ΔU = 0, according to ΔU = q + w, 0 = q + w so q = -w.

Edit: Forgot to put the - sign for the q = -w equation. Thanks for Justin Le for correcting me.
Last edited by Sungho Son 1G on Mon Jan 19, 2015 12:00 pm, edited 2 times in total.

Justin Le 2I
Posts: 142
Joined: Fri Sep 26, 2014 2:02 pm

### Re: Why internal energy is constant when temperature is cons

In an isothermal process $\Delta$U=q+p and since $\Delta$U=0, q=-w. The way of it is when work is being done on a system, the system must be losing heat at the same time and vice versa. Otherwise, the temperature and thus the internal energy would change.

Check out this link for more clarification:
viewtopic.php?f=128&t=1838

Neil DSilva 1L
Posts: 70
Joined: Fri Sep 26, 2014 2:02 pm

### Re: Why internal energy is constant when temperature is cons

I also found this (http://www.eoht.info/page/Joule%E2%80%99s+second+law) online about Joule's second law. It states that $U = f(T)$ (internal energy is a function of temperature). It follows the same logic as some of the other posts, if you feel like reading more into it.

Chem_Mod
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### Re: Why internal energy is constant when temperature is cons

In an ideal gas, particles are assumed to not interact with each other, therefore having no potential energy. So, the only energy left to contribute to the total is the kinetic energy. The temperature is a measure of the average kinetic energy. So, we can conceptually rationalize using the definition of temperature that dU=0 if dT=0.

The classical derivation of $\Delta U = 3/2 nRT$ is called the Kinetic Theory. It is more involved and uses some physics.

http://physics.bu.edu/~duffy/py105/Kinetictheory.html

However, for this class, it will be enough to just remember that Energy as well as Enthalpy are purely functions of temperature for ideal gases.