# Ideal gas law and thermodyanimic processes

1. The problem statement, all variables and given/known data
An ideal gas is enclosed in a cylinder with a movable piston at the top. The walls of the cylinder are insulated, so no heat can enter or exit. The gas initially occupies volume V1 and has pressure p1 and temperature T1. The piston is then moved very rapidly to a volume of V2=8.5V1. The process happens so rapidly that the enclosed gas does not do any work.

Find p2, T2, and the change in entropy of the gas. [Express your answers in terms of p1, T1, n, and R.]

2. Relevant equations
p1V1=p2V2
pV=nRT
ΔS=nRln(Vf/Vi)

3. The attempt at a solution

To determine p2 I used the relationship p1V1=p2V2

p2=(p1V1)/V2

p2=(p1V1)/(8.5V1)

p2=(p1)/8.5

To determine T2 we see that the process itself is an isothermal process since no work was done, and no heat escaped, Q=W. So the temperature will not have changed.
T2=T1

I can’t seem to determine ΔS correctly. Since the process is an isothermal expansion, the change in entropy is given by the equation

ΔS=nRln(V2/V1)

from this I can substitute 8.5V1 for V2, resulting in

ΔS=nRln((8.5)V1/V1)

ΔS=nRln(8.5)

This answer however, is incorrect. The problem asks to put in the answer in terms of p1, T1, n, R. The format of this question is a blank field that allows me to create an equation with subscripts, superscripts, fractions, matrices, etc. FYI.

Any help is appreciated, thanks in advance.

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