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I define the work done by the system as +dw and work done on the system as -dw.

Looking through my notes on Otto cycle, I observed that the for the two adiabatic curves in a clockwise cycle such that the direction on each curve is opposite to the other, the equation is given as (by the notes) du = Q - dw for an adiabatic compression and du = Q - dw for an adiabatic expansion.

Which doesn't make sense because one of curve if an adiabatic compression and the other is an adiabatic expansion.

Is there a mistake by the book?

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  • $\begingroup$ You've kinda already have the answer to this. The sign convention that you indicated is valid. The point is to remember that $dW$ can have a negative value. During compression, you have $dU$ = - $dW$, but $dW < 0$, so $dU$ is a positive quantity. $\endgroup$
    – user3823992
    Commented Apr 17, 2015 at 5:49

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It can be shown that the work done on an object follows the following equation:

$$ W = \int F\cdot ds = \int P\cdot dv$$

where $F$ is the force that acts upon the object, $ds$ is the distance it acts over, $P$ is the pressure acting on the object, $dv$ is the change in volume of the object, and the dot is the dot product. In thermodynamics, often when we talk of work we are talking about the work the gas does upon the environment, or the container used to hold it. The vector of the pressure will always be pointing outwards onto the container, and so therefore if the volume INCREASES, the work done is positive (change in volume and pressure in same direction) and if the volume DECREASES, the work done is negative.

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