Beyond Ideality

Question 1

The Ideal Gas Approximation

Answer 1

• the ideal gas equation makes the two approximations:
  1) particles don’t interact except through perfectly elastic collisions
  2) particles are treated as points that occupy zero volume

Question 2

Alternative Equations of State

van der Waals

Answer 2

P = RT/(V-b) - a/V²

Question 3

Alternative Equations of State

Dieterici

Answer 3

P = RTe^(-a/RTV) / (2V-b)

Question 4

Alternative Equations of State

Virial Expansion

Answer 4

P = RT/V {1 + B/V + C/V² + …}

-this increases in accuracy with more terms

Question 5

van der Waals Equation

Equation

Answer 5

P = nRT/(V - bn) + an²/V²

Question 6

The Helmholtz Equation

Answer 6

∂U/∂V |T = T * ∂P/∂T|V - P = T² ∂/∂T|V(P/T)

Question 7

The Helmholtz Equation

Proof

Answer 7

-starting from the first law
dU = dQ + dW
dU = TdS - PdV
-rearrange for dS = ...
dS = 1/T dU + P/T dV
-sub in dU = ∂U/∂V|T dV + ∂U/∂T|V dT
dS = 1/T(∂U/∂V||T + P)dV + 1/T ∂U/∂T|V dT
-at constant T, dT=0
∂S|T = 1/T(∂U/∂V|T + P) ∂V|T
-rearrange for ∂S/∂V|T
∂S/∂V|T = 1/T (∂U/∂V|T + P)
-using the Maxwell relation from dH, ∂S/∂V|T=∂P/∂T|V
∂P/∂T|V = 1/T (∂U/∂V|T + P)
-rearrange for ∂U/∂V = ...
∂U/∂V|T = T ∂P/∂T|V  -  P = T² ∂/∂T|v (P/T)

Question 8

Helmholtz Equation

For Ideal Gases

Answer 8

• sub in the ideal gas equation PV=nRT

- the Helmholtz equation reduces to 0

Question 9

Isothermal Compressibility

Answer 9

Κt = - 1/V ∂V/∂P|T

Question 10

Isothermal Compressibility

For Ideal Gases

Answer 10

Κt = 1/P

Question 11

Expansion Coefficient

Answer 11

α = 1/V ∂V/∂T|P

Question 12

Expansion Coefficient

For Ideal Gases

Answer 12

α = 1/T

Question 13

General Relation Between Cp and Cv

in terms of isothermal compressibility and expansion coefficient

Answer 13

Cp - Cv = Tα²V / Κt

-for an ideal gas this resduces to Cp - Cv = nR

Question 14

What does the internal energy of an ideal gas depend on?

Answer 14

• U of an ideal gas depends on T ONLY

- to prove this, sub in P=nRT/V to the Helmholtz equation

Question 15

When do the ideal gas assumptions break down?

Answer 15

-at high densities and low temperatures

Question 16

van der Waal’s Equation

Description

Answer 16

P = nRT/(V - bn) + an²/V²

Videal -> V - bn since the molecules themselves take up some volume
Pideal -> P + 𝛿P, where 𝛿P=a(N/V)² as the attractive force between molecules retards them by aN/V and the number density is N/V