An ideal-gas process is described by p=cV^ /2, where c is a constant. a.Find an expression for the work done on the gas in this process as the volume changes from V₁ to V₂.

Video transcript

Hi, everyone. In this practice problem, we're being asked to calculate the work done. We will have a guess following a process where non constant pressure follows the relation of B equals to A V squared A is the proportionality constant. And we're being asked to calculate the work done in moving the gas from V zero to TV zero. The options given for the work done as a nine A V zero cube, B 26 divided by two A V zero cube, C negative 13 A V zero cube and D negative 26 divided by three A V zero cube. So in this particular practice problem, we will assume that the ideal gas is going to be in question. So the gas is going to be ideal and the work done in the ideal gas process is going to be described by the relationship of W equals to negative integral of P multiplied by DV. It is given that P will be equals to A V squared or the relationship for P is given in the problem statement to be A V squared. So let's substitute that into our integration here. So W will be equals to negative integral from the initial volume which is P zero to the final volume, which is three P zero multiplied by A V squared multiplied by DV. That is essentially just substituting our pressure relation into our integration and integrating it from initial to final volume. Wanna recall the integration principle where the integral of X squared DX will be equal to one divided by three multiplied by X to the power of three or X cube plus C. So let's use that integration principle to actually calculate or integrate our equation that we have for the breakdown. So we will have the work or W to be equal to negative A multiplied by one divided by three P to the power of three. And we want to evaluate this from P 0 to 3 V zero. Evaluating our work done, we will then get work or W to then be equal to negative A divided by three multiplied by in parentheses, three P zero cube minus P zero cube, close parenthesis. And then simplify this. We will then get the word to then be equal to negative A divided by three multiplied by, in parentheses 27 P zero cube minus V zero cube which will then be equal to negative A divided by three multiplied by 26 P zero Q rearranging this so that it will be the same as our um answer choices. We will get the work done to then be equal to negative 26 divided by three multiplied by a multiplied by 30 Q. So that will be the final answer to this particular practice problem with the work done in moving the gas from P 0 to 3 P zero will be equal to negative point divided by three multiplied by a multiplied by P zero cube, which will correspond to option D in our answer choices. So option D will be the answer to this particular practice problem and that will be it for this video. If you guys still have any sort of confusion, please feel free to check out our adolescent videos on similar topics available on our website. But other than that, that will be it for this video. And thank you so much for watching.