Ideal gas law and thermodyanimic processes

[castrodisastro]

Homework Statement

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.]

Homework Equations

p₁V₁=p₂V₂
pV=nRT
ΔS=nRln(V_f/V_i)

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.

 

[Chestermiller]

Maybe they want you to put a number in for ln(8.50)?

 

[castrodisastro]

I tried putting

nR(2.14) but that was also incorrect

 

[Chestermiller]

I tried putting

nR(2.14) but that was also incorrect

I don't know what to say. That's the answer I would have obtained. Maybe 2.14nR, as if that could possibly matter.

 

[castrodisastro]

I am going to have to email the professor about it then. Thank you