Work done by ideal gas processes

[ahyaa]

Hello all,

In my physics textbook they discuss work done by ideal gas processes. The equation they give is W_gas = pΔV. I'm trying to figure it out if this is work done ON the gas by the surroundings, or work done BY the gas on surroundings.

From a previous chapter, they presented the conservation of energy equation as E_i + W_net,ext = E_f. Work in this equation is the net, external work on the system, such that +W is work by the surroundings on the system, and -W is work by the system on the surroundings. It would seem from this preliminary chapter that we would be defining any W as work ON the system by surroundings.

However W_gas can't possibly be work done ON the gas, because if a gas is expanding, it supposedly has +W_gas even though it is doing work on the surroundings (therefore W_net,ext should be negative). Could anybody confirm my thinking and does anybody know a good way to distinguish these two?

 

[DrClaude]

You pretty much have it. It all depends on the convention used, so if you take
$$
W_\mathrm{gas} = p \Delta V
$$
then $W_\mathrm{gas} > 0$ for work done by the gas (i.e., $\Delta V > 0$).

In the energy equation
$$
E_i + W_\mathrm{net,ext} = E_f
$$
$W_\mathrm{net,ext} > 0$ for work done on the gas. Therefore,
the total work will read
$$
W_\mathrm{net,ext} = -W_\mathrm{gas} + W_\mathrm{other}
$$

I must say that it is a strange convention: you usually would take the sign of $W$ to mean on/by directly. But it happens that depending on the situation, you would take one convention or the other depending on if the gas is the focus of attention or not. I know a textbook where the convention changes between two chapters, as it goes from looking at the properties of ideal gases to heat engines.

 

[Redbelly98]

Rather than keeping track of which convention is being used, I remember that work is positive when the force is exerted in the direction of motion. So for an expanding gas, the force exerted on the surroundings (by the gas) is in the direction of motion, while the force exerted on the gas (by the surroundings) is against the direction of motion. So, positive work is done on the surroundings, negative work is done on the gas.