### Delta S equation

Posted:

**Sat Jan 23, 2016 11:39 am**How is Delta S equal to nRln(V1/V2)?

Created by Dr. Laurence Lavelle

https://lavelle.chem.ucla.edu/forum/

https://lavelle.chem.ucla.edu/forum/viewtopic.php?f=133&t=10378

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Posted: **Sat Jan 23, 2016 11:39 am**

How is Delta S equal to nRln(V1/V2)?

Posted: **Sat Jan 23, 2016 2:27 pm**

Okay, so ΔS=. The type of expansion that occurs is isothermal, reversible expansion. Thus, since it's isothermal, ΔU=0, since ΔU=(3/2)*n*R*ΔT and ΔT=0, making ΔU=q+w=0 and q=-w. The work for an isothermal, reversible expansion is -nRTln(V2/V1), and since q(rev)=-w, q(rev)=-(-nRTln(V2/V1))=nRTln(V2/V1). Plugging this into the first equation, this makes ΔS==, and the temperature values cancel out to get nRln(V2/V1).

Posted: **Sat Jan 23, 2016 2:37 pm**

This question goes back to change in w=-p*change in V. Then you integrate both sides: w= -integral from V2 to V1 of (P*dV). You then substitute nRT/V for P and take the nRT out of the integral since they are constant. Then you integrate 1/V from V2 to V1 which simplifies into ln(V2/V1). In the end you should end up with, w=-nRT*ln(V2/V1). And when the change of internal energy equals 0, q=-w. and since Delta S=q/T, you can plug in the equation we just derived in for q. q=nRT*ln(V2/V1).

So, Delta S=(nRT*ln(V2/V1))/T. T then cancels and you are left with Delta S= nR*ln(V2/V1).

I hope that helped!

So, Delta S=(nRT*ln(V2/V1))/T. T then cancels and you are left with Delta S= nR*ln(V2/V1).

I hope that helped!