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Data Provided:

Chemistry Ancillary Booklet

THE UNIVERSITY OF SHEFFIELD

DEPARTMENT OF CHEMISTRY

Autumn Semester 2012/2013 2 hours duration

PHYSICAL CHEMISTRY

4 Questions – Answer ALL Questions.

Answer each QUESTION in a SEPARATE Answer Book.

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1. 

Answer all parts of this question.

(a)  Give a brief description of the difference between a first-order  and a second-order  

 phase transition.

Illustrate your answer diagrammatically.

[5 marks]

(b)  The Clapeyron equation for a liquid-vapour boundary is

vap

vap

 H dp

dT T V  

!=

!,

where the symbols have their usual meaning. Use this equation to estimate the

effect of raising the pressure on the boiling point of a typical liquid. The molar

volume of a perfect gas is 25 dm3 at 1 atm over the required temperature range.

[5 marks]

(c)  The Kelvin equation gives the vapour pressure, p, of a liquid when it is dispersed

as droplets of radius r :

2 mV rRT   p p e  ! "

=  

where p* is the bulk vapour pressure, and the other symbols have their usual

meaning.

Calculate the vapour pressure of a drop of water with radius 25 nm at 25°C, giventhat the vapour pressure of bulk water at this temperature is 5.623 kPa, and its

density is 994.0 kg m –3.

Assume a value of 72.0 mN m –1 for the surface tension, !.

[5 marks]

(d)  The Gibbs energy of one mole of a gas depends on the external pressure according

to the approximate equation

( ) ( ) ln   f  m f m ii

 pG p G p RT   p

! "= + # $% &

,

where the symbols have their usual meaning.

(i)  Derive this equation from first principles, stating any approximations made.

[3 marks]

(ii)  Calculate the change in chemical potential of a perfect gas when its pressure

is increased isothermally from 1.0 atm to 10.0 atm at 25°C.

[2 marks]

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2. 

Answer all parts of this question.

(a)  The time-independent Schrödinger equation for a single particle with one degree of

freedom and potential energy function V ( x) can be written in general form as

ˆ H E ! ! =  

(i) 

What do the symbols !  and E  represent here?

[2 marks]

(ii)  Give the detailed mathematical form of the Hamiltonian operator  in this one-

dimensional Schrödinger equation, defining all symbols that you introduce.

[2 marks]

(iii) 

What conditions must be obeyed by the functions ! (x) if they are to represent

 physically acceptable solutions for this problem?

[4 marks]

(iv) 

The position operator ˆ x and the momentum operator ˆ  x p do not commute.

What is the physical  consequence of this mathematical fact?

[1 mark]

(b)  A particular case of the one-dimensional Schrödinger equation applies to the

‘particle in a box’, where a particle is confined to the axis between x = 0 and x = L.

(i)  Describe and sketch the potential function V ( x) for this case.

[1 mark]

(ii) 

State the boundary conditions on the wavefunction for this case.

[1 mark]

(iii)  Describe briefly how application of the boundary conditions leads to theintroduction of a quantum number for the particle in a box.

[2 marks]

(iv) 

Give a formula for the energy levels of the particle-in-a-box system, definingall symbols that you introduce.

[2 marks]

(v)  Give a formula for the wavefunctions of the particle-in-a-box system, defining

all symbols that you introduce.

[2 marks]

(vi)  Given that the particle is an electron, and the box is of length 1 nm, calculate

the frequency of radiation needed to raise the system from its ground state to

its first excited state.

[3 marks]

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3. 

Answer all parts of this question.

Shown below is the rovibrational infrared absorption spectrum of carbon monoxide,12

C16

O (overview, top and detail of spectrum, bottom).

2050 2100 2150 2200

0.00

0.01

0.02

0.03

0.04

~

 

   l  n    (

        I   0

    /        I   )

!   / cm-1

2125 2130 2135 2140 2145 2150 2155 2160

0.00

0.01

0.02

0.03

0.04

R(3)

R(2)

R(1)

R(0)

P(4)

P(3)

P(2)

P(1)

~

 

   l  n 

   (        I   0

    /        I   )

!   / cm-1

 

(a)  Explain the meaning of " R(1)".

[1 mark]

(b)  What are the selection rules for electric dipole allowed infrared transitions

concerning the change in rotational quantum number  J ?

[2 marks]

(c)  Measure the positions of suitable lines, and determine the rotational constant B"

and B' of the ground state and of the excited state, respectively, using the method ofcombination differences. Use these values to calculate the bond length of carbon

monoxide in the ground state and in the excited state.

[Note that  B =

 BcI 

h2

8! 

 and  I B = µ   R2]

[10 marks]

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(d) 

In the vibrationally excited state, the rotational constant B' is expected to be slightly

lower than B" in the ground state. Why?

[2 marks]

(e)  Explain why the CO vibration can be observed by infrared spectroscopy, but the O2

vibration cannot.

[2 marks]

(f)  The stretching vibration of16

O2 occurs at 1556 cm-1

 and can be observed by Raman

spectroscopy. For Raman excitation light at "  = 532 nm, what is the wavelength (in

nm) of the red-shifted Raman scattered light?

[3 marks]

4.  Answer all parts of this question.

(a)  Answer all parts of this question.

A weak acid HA has an dissociation constant of 10 –3 M. 

(i)  Define the distribution ratio, D, for the acid and explain the difference

 between D and the distribution (or partition) coefficient K D.

(ii)  Calculate the percentage of acid left in the aqueous phase at pH 7, if 50 mL of

acid is extracted with 100 mL of dichloromethane (assume K D = 5).

(iii)  Explain how the distribution ratio of the acid changes between pH 2 and

 pH 7.

[5 marks]

(b)  Answer all parts of this question.

(i)  In HPLC, explain what is meant by gradient elution.

How does this differ from isocratic elution?

(ii)  Explain why gradient elution is preferable to an isocratic separation.

(iii) 

Would you increase or decrease the polarity of the mobile phase during

gradient elution when using a C18 stationary phase?

Justify your answer.

[5 marks]

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(c)  A sample containing a pesticide X  was analysed using Gas Chromatography (GC).

This sample was treated with an internal standard Q to give a concentration of

Q = 15.0 ppm.

A 1.0 µL sample injected into the GC, gives a peak area of 1012 for Q, and a peakarea of 3411 for X .

A 1.0 µL standard solution of 30.0 ppm Q with 15.0 ppm of pure X  was injectedgiving responses of 899 and 2791, respectively.

What is the concentration of X  in the sample?

[5 marks]

(d) 

Answer all parts of this question.

(i) 

Give a brief explanation of the basis for separation in ion-exchangechromatography (IEC).

(ii)  Describe the type of mobile phase composition that is needed to achieve

separation in IEC.

(iii) 

Describe the type of stationary phases that are needed to achieve separation in

IEC.

(iv) 

Describe two examples of analysis that are suitable for this kind of

chromatographic technique.

[5 marks]

END OF QUESTION PAPER