Methyl alcohol can be used as a fuel instead of, or combined with, gasoline. A sample of methyl alcohol, CH3OH, in a flask at constant volume exerts a pressure of 254 mm Hg at 57 degrees C. The flask is slowly cooled.

a) assuming no condensation, use the ideal gas law to calculate the pressure of the vapor at 35 degrees C.

Now I know using the first set of conditions I can find the moles and then do A that way, but I have a question first. When using the ideal gas law with the first set of conditions at constant volume, would I arbitrarily assign a volume to use in all calculations, or would I essentially leave it out of the ideal gas law?

Use an arbirtrary volume. Call it V, or if feel the urge, use 1 liter, that will but the moles on a per liter basis.

I have a follow-up question.
I calculated the number of moles (.01233) and plugged it into the ideal gas equation to find the pressure at 35 degrees C and 45 degrees C (.312 atm and .322 atm, consecutively). These were the pressures of the vapor.
The next questions state:
b) Compare your answers in a) with the equilibrium vapor pressures of methyl alcohol: 203 mmHg at 35 degrees C, and 325 mm Hg at 45 degrees C.
c) based on your answers from a) and b), predict the pressure exerted by the methyl alcohol in the flask at 35 degrees C, at 45 degrees C.
d) What physical states of methyl alcohol are present at both temperatures?

For b, I converted the mmHg pressures to atm (.2671 atm and .4276 atm consecutively), but I don't really get what the question is asking by "compare".
For c, I understand that I'm now supposed to find the pressure of the methyl alcohol, not the vapor as in a). But what is the difference in finding this? How does b) play a role in finding the pressure?
For d, is there an equation for this or should I just look it up?

For b, the question is asking you to compare the pressures you calculated in a) with the given equilibrium vapor pressures. Basically, you need to see how your calculated pressures compare to the equilibrium pressures, i.e., are they higher or lower?

For c, the equilibrium vapor pressure values given in b) indicate the pressure exerted by the methyl alcohol at these temperatures. So, the pressure you calculated in a) represents the total pressure exerted by the vapor, which includes both the methyl alcohol and any other gases present in the flask. Therefore, to find the pressure exerted by the methyl alcohol only, you should use the equilibrium vapor pressure values provided in part b) for 35 and 45 degrees C.

For d, you don't need an equation. You just need to think about what physical states (solid, liquid or gas) are possible for methyl alcohol at the temperatures given (35 and 45 degrees C). You can refer to the boiling point of methyl alcohol (64.7 degrees C) and the freezing point of methyl alcohol (-97.8 degrees C) to help you determine the physical states present at these temperatures. At 35 and 45 degrees C, it is likely that both liquid and vapor phases of methyl alcohol are present since both temperatures are between the freezing and boiling points.

For question b), when it asks you to compare your answers with the equilibrium vapor pressures of methyl alcohol, it means you need to compare the pressures you calculated in part a) with the given equilibrium vapor pressures at the same temperatures. You have already converted the given equilibrium vapor pressures from mmHg to atm, which is correct. Now, compare your calculated pressures to the equilibrium vapor pressures. If your calculated pressures are close to or approximately equal to the equilibrium vapor pressures, then it suggests that the system is approaching equilibrium, and the vapor pressure of methyl alcohol is consistent with its equilibrium conditions.

For question c), you are now asked to predict the pressure exerted by methyl alcohol in the flask at 35 degrees C and 45 degrees C. Remember that your previous calculations in part a) were for the pressure of the vapor. To find the pressure exerted by methyl alcohol in the flask, you need to subtract the pressure of the vapor from the total pressure inside the flask.

Total pressure inside the flask = Pressure of the vapor + Pressure exerted by methyl alcohol.

Given that you have already calculated the pressure of the vapor (from part a)) and you also have the total pressure (254 mmHg at 57 degrees C), you can find the pressure exerted by methyl alcohol using the following equation:

Pressure exerted by methyl alcohol = Total pressure - Pressure of the vapor.

For part d), you can determine the physical states of methyl alcohol at both temperatures by comparing the calculated pressure of the vapor with the given equilibrium vapor pressures. If the pressure of the vapor is higher than the equilibrium vapor pressure, it suggests that the methyl alcohol is in the gaseous state. If the pressure of the vapor is lower than the equilibrium vapor pressure, it implies that the methyl alcohol is in the liquid state. If the pressure of the vapor is equal to the equilibrium vapor pressure, then it suggests that the system is in equilibrium, with both liquid and vapor states coexisting.