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1 t = 1 t a + 1 t b size 12{ { {1} over {t} } = { {1} over {t rSub { size 8{a} } } } + { {1} over {t rSub { size 8{b} } } } } {}

For reference, the exact lineshape function (assuming two equivalent groups being exchanged) is given by the Bloch Equation, [link] , where g is the intensity at frequency v , and where K is a normalization constant

g ( v ) = Kt ( v a + v b ) 2 [ 0 . 5 ( v a + v b ) v ] 2 + 2 t 2 ( v a v ) 2 ( v b v ) 2 size 12{g \( v \) = { { ital "Kt" \( v rSub { size 8{a} } +v rSub { size 8{b} } \) rSup { size 8{2} } } over { \[ 0 "." 5 \( v rSub { size 8{a} } +v rSub { size 8{b} } \) -v \] rSup { size 8{2} } +4p rSup { size 8{2} } t rSup { size 8{2} } \( v rSub { size 8{a} } -v \) rSup { size 8{2} } \( v rSub { size 8{b} } -v \) rSup { size 8{2} } } } } {}

Low temperatures to coalescence temperature

At low temperature (slow exchange), the spectrum has two peaks and Δ v >>t. As a result, [link] reduces to [link] , where T 2a’ is the spin-spin relaxation time. The linewidth of the peak for species a is defined by [link] .

g ( v ) a = g ( v ) b = KT 2a 1 + T 2a 2 ( v a v ) 2 size 12{g \( v \) rSub { size 8{a} } =g \( v \) rSub { size 8{b} } = { { ital "KT" rSub { size 8{2a} } } over {1+T rSub { size 8{2a} rSup { size 8{2} } } \( v rSub { size 8{a} } -v \) rSup { size 8{2} } } } } {}
( Δv a ) 1 / 2 = 1 π ( 1 T 2a + 1 t a ) size 12{ \( Dv rSub { size 8{a} } \) rSub { size 8{1/2} } = { {1} over {p} } \( { {1} over {T rSub { size 8{2a} } } } + { {1} over {t rSub { size 8{a} } } } \) } {}

Because the spin-spin relaxation time is difficult to determine, especially in inhomogeneous environments, rate constants at higher temperatures but before coalescence are preferable and more reliable.

The rate constant k can then be determined by comparing the linewidth of a peak with no exchange (low temp) with the linewidth of the peak with little exchange using [link] , where subscript e refers to the peak in the slightly higher temperature spectrum and subscript 0 refers to the peak in the no exchange spectrum.

k = π 2 [ ( Δv e ) 1 / 2 ( Δv 0 ) 1 / 2 ] size 12{k= { {p} over { sqrt {2} } } \[ \( Dv rSub { size 8{e} } \) rSub { size 8{1/2} } - \( Dv rSub { size 8{0} } \) rSub { size 8{1/2} } \] } {}

Additionally, k can be determined from the difference in frequency (chemical shift) using [link] , where Δ v 0 is the chemical shift difference in Hz at the no exchange temperature and Δ v e is the chemical shift difference at the exchange temperature.

k = π 2 ( Δv 0 2 Δv e 2 ) size 12{k= { {p} over { sqrt {2} } } \( Dv rSub { size 8{0} rSup { size 8{2} } } -Dv rSub { size 8{e} rSup { size 8{2} } } \) } {}

The intensity ratio method, [link] , can be used to determine the rate constant for spectra whose peaks have begun to merge, where r is the ratio between the maximum intensity and the minimum intensity, of the merging peaks, I max /I min

k = π 2 ( r + ( r 2 r ) 1 / 2 ) 1 / 2 size 12{k= { {p} over { sqrt {2} } } \( r+ \( r rSup { size 8{2} } -r \) rSup { size 8{1/2} } \) rSup { size 8{-1/2} } } {}

As mentioned earlier, the coalescence temperature, T c is the temperature at which the two peaks corresponding to the interchanging groups merge into one broad peak and [link] may be used to calculate the rate at coalescence.

k = πΔv 0 2 size 12{k= { {pDv rSub { size 8{0} } } over { sqrt {2} } } } {}

Higher temperatures

Beyond the coalescence temperature, interchange is so rapid (k>>t) that the spectrometer registers the two groups as equivalent and as one peak. At temperatures greater than that of coalescence, the lineshape equation reduces to [link] .

g ( v ) = KT 2 [ 1 + πT 2 ( v a + v b 2v ) 2 ] size 12{g \( v \) = { { ital "KT" rSub { size 8{2} } } over { \[ 1+pT rSub { size 8{2} } \( v rSub { size 8{a} } +v rSub { size 8{b} } -2v \) rSup { size 8{2} } \] } } } {}

As mentioned earlier, determination of T 2 is very time consuming and often unreliable due to inhomogeneity of the sample and of the magnetic field. The following approximation ( [link] ) applies to spectra whose signal has not completely fallen (in their coalescence).

k = 0 . Δv 2 ( Δv e ) 1 / 2 ( Δv 0 ) 1 / 2 size 12{k= { {0 "." 5pDv rSup { size 8{2} } } over { \( Dv rSub { size 8{e} } \) rSub { size 8{1/2} } - \( Dv rSub { size 8{0} } \) rSub { size 8{1/2} } } } } {}

Now that the rate constants have been extracted from the spectra, energetic parameters may now be calculated. For a rough measure of the activation parameters, only the spectra at no exchange and coalescence are needed. The coalescence temperature is determined from the NMR experiment, and the rate of exchange at coalescence is given by [link] . The activation parameters can then be determined from the Eyring equation ( [link] ), where k B is the Boltzmann constant, and where ΔH - TΔS = ΔG .

ln ( k T ) = ΔH RT ΔS R + ln ( k B h ) size 12{"ln" \( { {k} over {T} } \) = { {DH rSup { size 8{³} } } over { ital "RT"} } - { {DS rSup { size 8{³} } } over {R} } +"ln" \( { {k rSub { size 8{B} } } over {h} } \) } {}

For more accurate calculations of the energetics, the rates at different temperatures need to be obtained. A plot of ln(k/T) versus 1/T (where T is the temperature at which the spectrum was taken) will yield ΔH , ΔS , and ΔG . For a pictorial representation of these concepts, see [link] .

Questions & Answers

anyone know any internet site where one can find nanotechnology papers?
Damian Reply
research.net
kanaga
Introduction about quantum dots in nanotechnology
Praveena Reply
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
Damian Reply
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
Akash Reply
it is a goid question and i want to know the answer as well
Maciej
characteristics of micro business
Abigail
for teaching engĺish at school how nano technology help us
Anassong
Do somebody tell me a best nano engineering book for beginners?
s. Reply
there is no specific books for beginners but there is book called principle of nanotechnology
NANO
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
s.
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
Tarell
what is the actual application of fullerenes nowadays?
Damian
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
Tarell
what is the Synthesis, properties,and applications of carbon nano chemistry
Abhijith Reply
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
NANO
so some one know about replacing silicon atom with phosphorous in semiconductors device?
s. Reply
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Harper
Do you know which machine is used to that process?
s.
how to fabricate graphene ink ?
SUYASH Reply
for screen printed electrodes ?
SUYASH
What is lattice structure?
s. Reply
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
On having this app for quite a bit time, Haven't realised there's a chat room in it.
Cied
what is biological synthesis of nanoparticles
Sanket Reply
what's the easiest and fastest way to the synthesize AgNP?
Damian Reply
China
Cied
types of nano material
abeetha Reply
I start with an easy one. carbon nanotubes woven into a long filament like a string
Porter
many many of nanotubes
Porter
what is the k.e before it land
Yasmin
what is the function of carbon nanotubes?
Cesar
I'm interested in nanotube
Uday
what is nanomaterials​ and their applications of sensors.
Ramkumar Reply
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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Good
advantages of NAA
Sai Reply
how I can reaction of mercury?
Sham Reply

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Source:  OpenStax, Physical methods in chemistry and nano science. OpenStax CNX. May 05, 2015 Download for free at http://legacy.cnx.org/content/col10699/1.21
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