# Entropy calculation

## Homework Statement

A thermally insulated cylinder, closed at both ends, is fitted with a frictionless heat-conducting piston that divides the cylinder into two parts. Initially, the piston is clamped in the center with 1 liter of air at 300 K and 2 atm pressure on one side and 1 liter of air at 300K at 1 atm pressure on the other side. The piston is released and reaches equilibrium in pressure and temperature at a new position. Compute the final pressure and temperature and increase of entropy if air is assumed to be the ideal gas. What irreversible process has taken place?

## Homework Equations

$$\Delta$$S = $$\Delta$$Q/T

## The Attempt at a Solution

This is an isothermal free expansion. So temperature remains constant. I know the final pressure will be 1.5 atm (intuitively, (1+2)/2), but how can I compute this out by ideal gas law?

By PV=nRT,
(101325)(1000/1000000)=n (8.31)(300)
n1(the compartment with 1atm)=0.040643802
n2(the compartment with 2atm) = 0.081828
TOTAL number of moles = 0.12193

When I search through the Internet, I find this equation
$$\Delta$$S=nRln(Vf/Vi) = 0.12193(8.31)ln2 = 0.702323.........

but the answer is 0.0566, what's wrong with my calculation???

Mapes
Homework Helper
Gold Member
Why did you use $V_f/V_i=2$? That ratio is supposed to represent the final and initial volumes.

Actually, from the phase'1 liter of air', i am not quite sure about it is about the volume or number of moles. Moreover, when I think this question again, it is not free expansion as the sentence "The piston is released and reaches equilibrium in pressure and temperature at a new position." provides the fact that the piston is still in the cylinder. Then another question comes, if it is not free expansion, then, what irreversible process has taken place.

Together with this question, i have one more question to ask...
Answer: 1.5 atm. 300K, 0.0566 J/K
For the first answer, the piston stops moving when the pressure of the two compartments are equal, that is 1.5 atm.
From the second answer, it seems that the temperature is assumed to be unchanged. However, we know that PV=nRT, it may be the case that the temperature increase.
Anybody can help me to clarify? It may be crucial in solving the increase in entropy.
Thanks You.

Mapes
Homework Helper
Gold Member
Actually, from the phase'1 liter of air', i am not quite sure about it is about the volume or number of moles.

The liter is a measure of volume. By applying an assumption (ideality), and knowing the pressure and temperature, one can calculate the amount of gas in moles.

Moreover, when I think this question again, it is not free expansion as the sentence "The piston is released and reaches equilibrium in pressure and temperature at a new position." provides the fact that the piston is still in the cylinder. Then another question comes, if it is not free expansion, then, what irreversible process has taken place.

It is not a free expansion (because the gas does not expand into a vacuum), but it is a spontaneous expansion.

From the second answer, it seems that the temperature is assumed to be unchanged. However, we know that PV=nRT, it may be the case that the temperature increase.
Anybody can help me to clarify? It may be crucial in solving the increase in entropy.
Thanks You.

Can the gases end up at different temperatures, considering the boundary between them?
Can the gases end up at an average temperature different from 300K, considering the boundary between the gases and the outside environment?

I can figure out the calculation of the increase in entropy now. However, I still cannot understand why the temperatures remain unchanged. Thank You.

Mapes