Calculate the Enthropy Change of Isothermal Expansion of a Non-Ideal Gas

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SUMMARY

The discussion focuses on calculating the entropy change (ΔS) during the isothermal expansion of a non-ideal gas using the formula ΔS = nR ln(V2/V1) for part (a), where V2 is 20L and V1 is 2L, resulting in ΔS = 19.15 J/K. In part (b), the user encounters difficulties in solving for pressure (P) using the van der Waals equation, P = [RT/(v-b)] - [a/v^2], and seeks alternative methods beyond Newton-Raphson for further calculations. Part (c) confirms the integration approach for ΔS and explores the relationship between pressure and temperature through Maxwell's relations.

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  • Understanding of thermodynamic concepts, particularly entropy and isothermal processes.
  • Familiarity with the van der Waals equation for non-ideal gases.
  • Knowledge of integration techniques in calculus.
  • Basic understanding of Maxwell's relations in thermodynamics.
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  • Learn about alternative numerical methods for solving equations, such as the bisection method or secant method.
  • Explore the derivation and application of Maxwell's relations in thermodynamic systems.
  • Investigate the implications of non-ideal gas behavior on entropy calculations in various thermodynamic processes.
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Homework Statement
a) Calculate the entropy change for an isothermal expansion of a Van Der Waals Gas from V1 to V2. b) Use this to calculate ΔS for 1 mole of NH3 from 2L to 20L at 298K. c) Compare this to an Ideal Gas
Relevant Equations
ΔS = ∫ (dq/T)
PV=nRT Ideal Gas
[P + a/(v/n)[SUP]2[/SUP]][v/n - b] = RT Not Ideal
Part (a)
ΔS = ∫ (dq/T)

because: dq = PdV = (nRT/V)dV

Then:
ΔS = ∫ (1/T)*(nRT/V)dV
ΔS = nR ∫(1/V) dV
ΔS =nR[ln(V2/V1)]

Part (b)
This is where I'm stuck. I know [P + a/(v/n)2][v/n - b] = RT can be solved for P and simplified to
P = [RT/(v-b)]-[a/v2] since n=1mol

But I don't know how to proceed from here, to solve for v and in turn solve ΔS. I can't use Boyle's Law and I was told there's another method besides Newton-Rapson.

Part (c)
Take ΔS =nR(ln(V2/V1) from part (a) and input: V2=20L |V1=2L

ΔS = (1mol)(8.3145 J*mol/K)(ln(10))=19.15 J/K

19.15J/K - (part b) = Final Answer
 
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I think I got it:
P = [RT/(v-b)]-[a/v2]

Integrate with respect to T:

dP/dT = RT/[(v/n)-b] (d/dT) - [a/v2] (d/dT)

[a/v2] (d/dT) goes to 0 so

dP/dT = R/[(v/n)-b]

And a Maxwell states: dP/dT = dS/dV

so: dS/dV = R/[(v/n)-b]

dS = R/[(v/n)-b] dV

ΔS = ∫ R/[(v/n)-b] dV

And then just integrate from 2L to 20L? Is this correct for Part b?
 

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