n moles of an ideal gas at temperature T1 and volume V1 expand isothermally until the volume has doubled. In terms of n, T₁ , and V₁, what are (c) the heat energy transferred to the gas?

Video transcript

Hi, everyone. In this practice problem, we're being asked to determine the amount of heat energy transfer. We will have an ideal gas used as fuel in a high tech engine where the engine starts with M moles of the gas at an initial temperature of T I and a volume of V I. And during the engine's operation, the gas undergoes an isothermal expansion process, maintaining a constant temperature. While the volume triples, we're being asked to determine the amount of heat energy transferred to the guest during this process. In terms of MT I and V I, the options given are am RT Iln three B negative M RT Iln two C negative mtiln three and dmtiln two. So, since the gas undergoes an isothermal process, the internal energy of the gas will not change. Thus, we will apply the first law of thermodynamics. So the first law of thermo will then be delta EPH will be equal to W plus Q which will be equal to zero. Where in this case, W is the work done on the gas and Q is the heat energy transfer to the gas during this process. And from this equation for the first law of thermodynamics, we can then rearrange so that Q will then be equal to negative W. So Q is the one that we are interested to find and that will be equal to negative W based on the first law of thermodynamics. Now, we want to find the value of W and the work done in an isothermal process is going to be given by W equals to negative M multiplied by R multiplied by T multiplied by the LNFVF divided by V I. So in this case, M is the number of moles R is the ideal gas constant D is the temperature. VF is the final volume and V I is the initial volume. So since in the problem statement, it is given that the final volume is triple, so VF will then be equal to three V I. Let's substitute that into our W FI formula here. So W will be equal to negative M RT multiplied by LNRVF will be three vim uh divided by V I. We can cross up the V I so that the W will then be equal to negative M RT multiplied by Ln of three. So we want to substitute the value of W in the equation that we have from the first law of thermodynamics so that we will get Q to then be equal to negative of the negative M RT multiplied by Ln of three just like. So the T here is going to be P I and we can substitute that and simplify our Q equation or the amount of heat energy transferred to the gas so that the Q will be equal to M multiplied by R multiplied by T I multiplied by Ln of three. So that will be the final equation or the final answer to this particular practice problem with the amount of heat energy transferred to the guest during the process will be equal to M multiplied by R multiplied by T I multiplied by Ln of three, which will correspond to option A in our answer choices. So option A 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'll be it for this one and thank you so much for watching. Bye bye.