4a: Solutions and Mixtures

Question 1

What is the equation for internal energy - the first law of thermodynamics?

Answer 1

dU = dQ - PdV

Question 2

What is the equation for entropy - the second law of thermodynamics?

Answer 2

dS = dQ/T

Question 3

How do you cancel variables to change thermodynamic functions?

Answer 3

The product rule - add the differential required to change the function and use the product rule to change the function by cancelling a variable.

dU = TdS - PdV

dU + d(PV) = TdS + VdP

dH = TdS + VdP

Question 4

Define chemical potential in words and by equation.

Answer 4

The free energy per number of molecules

μ = (∂G/∂N)_T,P

G = Nμ

Question 5

Define extensive and intensive quantities and sort the quantities into each.

Answer 5

Extensive - scale with the system: U, S, V, N

Intensive - independant of the system: T, P

Question 6

Define the molar quantities for a pure substance.

Answer 6

g = G/N = μ

s = S/N

h = H/N

v = V/N

Question 7

Find the Gibbs-Dunhem equation from G = H - TS.

(note that dH = dU + d(PV) = TdS + VdP + μdN)

Answer 7

G = H - TS (differentiate)

dG = dH - d(TS) = dU + d(PV) - d(TS)

dG = TdS - PdV + d(PV) + μdN - d(TS)

dG = -SdT + VdP + μdN

dG = -SdT + μdN + VdP (combine with dG = μdN + Ndμ)

Ndμ = -SdT + VdP

Question 8

How does chemical potential change when a particle changes state while the states are in equlibrium? What is the effect of this?

Answer 8

The chemical potential of both states is equal, μ^I = μ^II

This means μ = Δh - TΔs goes to Δh = TΔs.

Question 9

How do you get enthalpy change from a phase diagram?

Answer 9

The Clapeyron equation:

dP/dT = Δs/Δv = Δh/TΔv

dP/dT is the gradient of the phase diagram.

Question 10

What is the application of the Clausius-Clapeyron equation?

Answer 10

On the phase boundary between a gas and a liquid/solid:

dP/dT = PΔs/RT = PΔh/RT²

Question 11

How can the Clausius-Clapeyron equation be linearized?

Answer 11

Change the variables on the phase boundary to lnP and 1/T, the gradient, dlnP/dT^-1 = -Δh/R

Question 12

How is the Gibbs-Duhem equation change when it is applied to mixtures?

Answer 12

Σ_iN_idμ_i = -SdT + VdP

The i is the species index and represents each different component of the mixture.

Question 13

What is the equation for boiling point elevation?

Answer 13

ΔT_b = K_bm₂

m₂ = solute molality

K_b = ebullioscopic constant

Question 14

Define molality.

Answer 14

The amount of solute in moles divided by the mass of solvent in kg.

Question 15

What is the equation for freezing point depression?

Answer 15

-ΔT_f = K_fm₂

K_f = cryoscopic constant

Question 16

What is the equation for osmotic pressure?

Answer 16

Π = RTc₂

c₂ = molarity of the solute

Question 17

How is vapour pressure related to solution affinity? How is this affected in perfect and ideal solutions?

Answer 17

a_i = P_i/P⁰_i

Activity = vapour pressure/vapour pressure of pure solvent

Vapour pressure increases as the affinity of the solute in the solution decreases.

In a perfect solution (hexane in heptane would be close) the acitivity is equal to the mole fraction of the solution.

In reality, ideal solutions only follow this trend in dilute solutions, this is Raoult’s law.

The vapour pressure of the solvent is proportional to the mole fraction of the solvent.

Question 18

What is Henry’s law for solute?

Answer 18

a₂ ∝ x₂ when x₂ << 1

P₂ = k_Hx₂

Where x₂ = solubility

The solubility is proportional to the partial pressure of the solute in the gas phase.

Question 19

How do you find if a solution deviates from ideality (similar solution and solute)?

Answer 19

The activity coefficient, γ_i = a_i/x_i

γ > 1: weaker affinity to solution

γ < 1: stronger affinity to solution

Where a_i = P_i/P⁰_i

x_i = mole fraction

P = vapour pressure

Question 20

How do you work out what proportion of different phases there is present in a seperated mix of liquids?

Answer 20

Measure the length of the lines between their minimums, the further from the minimum point, the smaller the ratio between the mixes.

Question 21

What is the Gibbs phase rule?

Answer 21

F = C - P + 2

F = degrees of freedom

C = number of components

P = number of phases

Question 22

Define the free energy of mixing by words and equation.

Answer 22

The free energy difference between a solution mixtures and pure solute and solvent.

g^mix = RT(x₁lna₁ + x₂lna₂)

Question 23

What is Raoults law for solvent?

Answer 23

When the solute is dilute, the vapour pressure of the solvent, P1, is proportional to the mole fraction of the solvent, x1.

P₁ = P₁⁰x₁