Calculate Piston Height in Ideal Gas Law Problem | Homework Statement

[jthekk2]

Homework Statement

A vertical cylinder has a piston of mass "m" on top that is free to move without friction. If
there are "n" moles of an ideal gas in the cylinder at absolute temperature "T", what is the
height "h" of the piston above the bottom of the cylinder so that it will be in equilibrium
under its own weight?

Homework Equations

PV=nRT
P=F/A
V=A*h

The Attempt at a Solution

I worked this through summing the forces on the piston and got F_gas - F_atm -mg =0 or P_gas*A - P_atm*A = mg

this gives P_gas = mg/A + P_atm

from here I looked at the Ideal Gas Law Equation,

PV=nRT, solving for V and substitution we get

A*h = nRT/(P_gas)

so h = nRT/(P_gas*A)

plugging in the P_gas I got:

h = nRT/(mg + P_atm*A)

The thing is, the area was not mentioned in the question and I don't believe we're supposed to have area in the final answer (which is in variable form). I know the work I did above is correct, but is there a way to get rid of the A, as I am 90% sure it isn't supposed to be there. Thanks!

 

[anathema]

Thank you for your question! Your approach to solving this problem is correct so far. However, I can understand your concern about the area not being mentioned in the question and possibly not being needed in the final answer.

In this case, we can assume that the area of the piston is equal to the area of the cylinder base. This assumption is based on the fact that the piston is said to be "on top" of the cylinder and is free to move without friction. This implies that the piston is in direct contact with the gas inside the cylinder, and therefore must have the same area as the base of the cylinder.

Assuming this, we can substitute A = πr^2 into your final equation, where r is the radius of the cylinder base. This will give us:

h = nRT/(mg + P_atm * πr^2)

This final equation does not have the area (A) variable in it, and therefore can be used as the solution to the problem. I hope this helps! Let me know if you have any further questions.

 

[nlightner]

I would suggest double-checking the given information and problem statement to ensure that there is no missing information or errors. If the problem statement is indeed correct and there is no mention of the area, then the final answer should not contain the variable A.

One way to eliminate the variable A is to use the ideal gas law equation in terms of density rather than pressure. This would give us:

PV = nRT
m/ρ = nRT
ρ = m/(nRT)

where ρ is the density of the gas. We can then substitute this into the equation for the height of the piston:

h = nRT/(mg + P_atm*A)
h = nRT/(mg + P_atm*A) * (nRT/m)
h = nRT/(ρg + P_atm*A)

This eliminates the A variable and gives us the height of the piston in terms of the gas density.