**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

**V**and has pressure

_{1}**p**and temperature

_{1}**T**. The piston is then moved very rapidly to a volume of

_{1}**V**. The process happens so rapidly that the enclosed gas does not do any work.

_{2}=8.5V_{1}Find **p _{2}**,

**T**, and the change in entropy of the gas. [Express your answers in terms of

_{2}**p**,

_{1}**T**,

_{1}**n**, and

**R**.]

**2. Relevant equations**

**p _{1}V_{1}=p_{2}V_{2}**

pV=nRT

ΔS=nRln(V_{f}/V_{i})

**3. The attempt at a solution**

To determine **p _{2}** I used the relationship

**p**

_{1}V_{1}=p_{2}V_{2}**p _{2}=(p_{1}V_{1})/V_{2}**

**p _{2}=(p_{1}V_{1})/(8.5V_{1})**

**p _{2}=(p_{1})/8.5**

To determine **T _{2}** 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.

**T**

_{2}=T_{1}
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(V _{2}/V_{1})**

from this I can substitute **8.5V _{1}** for

**V**, resulting in

_{2}**ΔS=nRln((8.5)V _{1}/V_{1})**

**ΔS=nRln(8.5)**

This answer however, is incorrect. The problem asks to put in the answer in terms of **p _{1}**,

**T**,

_{1}**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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