Question

In an adiobatic system, what's the relationship between heat and change in internal energy?

Q=0 in an adiobatic system because it's an insulated system (let's imagine an ideal system) where there's no temperature change. If change in internal energy is always equal to `n*Cv*dT` and there's no temperature change, then there shouldn't be a change in internal energy? But this wouldn't make sense because the system would overall just be undefined or something.

Someone please help! Thank you

Answer 1

Heat `!=` temperature.

Explanation:

The important point to realize is that an adiabatic process does not imply that there is no temperature change. An isothermal process is one in which there is no temperature change. A process is adiabatic if it occurs so quickly (or so slowly) that the energy input (into the system) by heating is zero, or in other words, no heat is lost or gained by the system.

There is a component that you are forgetting: work.

• The first law of thermodynamics states:

`color(blue)(Q=W+DeltaE)`

where `Q` is the energy input by heating, `W` is the work done, and `DeltaE` is the change in internal energy

• When `Q=0`, we have:

`color(blue)(0=W+DeltaE)`

Which tells us that the work done (by system or environment depending on how you structure your signs), is equal to the change in internal energy.

Let's for example consider an adiabatic expansion of a gas. Here, the gas does work on the environment as it expands, so `W` is a positive quantity. Then:

`W=-DeltaE`

We know that the work is positive so we must have a negative change in internal energy (to cancel the negative), which implies that `E_f-E_i` is a negative quantity. This tells us that the initial energy of the system was greater than the final energy.

• One way to define the internal energy (average translational kinetic energy of `N` molecules/atoms/particles):

`color(blue)(E=3/2Nk_bT)`

where `k_b` is Boltzmann's constant and `T` is the temperature of the system

• Because the number of molecules/particles/etc. remains constant and `k_b` is by definition a constant, the only factor here which can change is the temperature.

• We must have the the internal energy decreases in this adiabatic process. It does not remain constant.