91Ó°ÊÓ

For an adiabatic process in an ideal gas, particularly compression (for which work done is negative), the formula for work done is the following: \( W_{process} = - \frac{(\gamma - 1)}{\gamma} P_i V_i \left(1 - \left(\frac{V_f}{V_i}\right)^{(\gamma-1)}\right) \) Here, \(W_{process}\) is the work done by the gas, \(\gamma\) is the adiabatic index (the ratio of specific heats, which varies for different gases), \(P_i\) and \(V_i\) are the initial pressure and volume of the gas, and \(V_f\) is the final volume of the gas. However, we cannot obtain the specific numbers to plug into the formula with the given exercise. The student should recognize that they only need to know the fraction in the formula that corresponds to work done.

In each option, the given work done is written in the form: \(W = - \frac{X}{32} RT\), where X is a constant. Note that the relationship between work done and the initial pressure and volume is not essential in this case. We only need the fraction that relates the work done to the gas constant R and temperature T. Rewrite the adiabatic work done formula as: \( \frac{W_{process}}{-P_i V_i} = \frac{(\gamma - 1)}{\gamma} \left(1 - \left(\frac{V_f}{V_i}\right)^{(\gamma-1)}\right) \) The term on the left-hand side of the equation is a fraction representing the relationship between work done and initial pressure and volume. Since we are looking for the fraction corresponding to work done, gas constant, and temperature, we can make an assumption that the given options are simplified forms of the above formula. Now, it is up to the student to recognize the formula they have previously learned, match it with one of the given options, and choose the correct answer. In conclusion, since the exercise does not provide any specific information besides the potential formulae for work done, the student should have prior knowledge of the formula they have learned and choose the correct answer based on that knowledge.