# What are the signs for work done by (on?) a gas during isothermal reversible compression?

For the expansion of a gas, isothermally and reversibly, what is the sign taken? Do I use this equation $-nRT\ln (Vf/Vi)$ or this one $nRT\ln (Vf/Vi)$ ? Moreover, when I'm asked about the 'work done', what exactly does the work? The gas itself? The surroundings? I'm confused about this because my text says one thing and the guidebook, another. Thanks.

The easiest way to remember the sign conventions is to think of the gas particles as "wanting" to get away from each other. To reduce the volume of the gas, you have to do work on the gas - the surroundings are doing the work. In chemistry we always take the point of view of the system. If we do work on the system, we add energy to it, and therefore, the sign of work is positive.

This means the opposite is true as well. When the volume of the system increases, the sign of the work term must be negative - the system is losing energy and doing work on the surroundings.

For your equations, all of this holds for the first one:

$$-nRT\ln (Vf/Vi)$$

This is because a decrease in volume will give you a fraction in the $\rm{ln}$ term, which leads to a negative sign. The leading negative sign reverses that. The result is that any compression will have a positive work term, and any expansion will have a negative work term.

You can see how the correct sign is a result of the derivation in the wikipedia article.

• THANKS!!! Is the physical or chemical view? Moreover, during the expansion of a gas, is the work done by the gas? If so, is it positive? My textbook tells me to calculate merely 'work done' at constant external pressure and ends up with +16.1 L atm. What does this work? – Christopher Mar 9 '15 at 15:05
• Work is a transfer of energy: E = q + w. This means that work done on the system is adding energy to the system, so the sign must be positive. During expansion, the system (the gas) is doing work on the surroundings, so the sign should be negative. If they have a positive sign for work during an expansion, then they are taking the opposite convention, or asking from the point of view of the surroundings. – thomij Mar 9 '15 at 15:18
• I see, so essentially. In the case given here (this is Chemistry not Physics by the way), they're asking for the work done 'on' the system. The case in topic, to be specific is here, and the reason I ask is in lieu of my prime examination tomorrow. imgur.com/jKGvHRN – Christopher Mar 9 '15 at 15:23
• I would check with whomever is grading the exam. In the classes I teach, if the question is worded ambiguously, I tell my students to write something like "sign is negative because the system is doing the work" - I don't care if they can guess what I meant in the question, what I care about is whether or not they understand what the signs mean and what the direction of energy transfer is. Your examiner might care about something different. – thomij Mar 9 '15 at 17:11
• Shouldn't it be "when the volume of the system increases" instead? – Soham Chowdhury Mar 17 '18 at 10:07