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Summation Formula: 

Constant Series

\begin{equation}
\sum_{k=0}^{S} c=s c
\end{equation}

Example: If S = 4

\begin{equation}
\begin{array}{c}{\sum_{k=0}^{4} c=0+c+c+c+c=4 c} \\ {\sum_{k=0}^{4} c=4 c}\end{array}
\end{equation}

Linear Series: \begin{equation}
\sum_{k=0}^{S} k=\frac{S(S+1)}{2}
\end{equation}

Example: If S = 4 \begin{equation}
\begin{array}{c}{\sum_{k=0}^{4} k=0+1+2+3+4=10} \\ {\sum_{k=0}^{S} k=\frac{S(S+1)}{2}=10}\end{array}
\end{equation}

Quadratic Series

\begin{equation}
\sum_{k=0}^{S} k^{2}=\frac{S(S+1)(2 S+1)}{6}
\end{equation}

Example if S = 4 \begin{equation}
\begin{array}{l}{0^{2}+1^{2}+2^{2}+3^{2}+4^{2}=0+1+4+9+16=30} \\ {\sum_{k=0}^{4} k^{2}=\frac{4(4+1)(2 \times 4+1)}{6}=\frac{4 \times 5 \times 9}{6}=30}\end{array}
\end{equation}

Roos of Quadratic Equation

If, \begin{equation}
a x^{2}+b x+c=0
\end{equation}

then, \begin{equation}
x=\frac{-b \pm \sqrt{b^{2}-4 a c}}{2 a}
\end{equation}

Binomial Expansion

\begin{equation}
(1+x)^{n}=1+n x+\frac{n(n-1)}{2 !} x^{2}+\frac{n(n-1)(n-2)}{3 !} x^{3}+\ldots \ldots \ldots \ldots \ldots
\end{equation}

To compute \((a+b)^{n}\)

\begin{equation}
(a+b)^{n}=a^{n}(1+x)^{n}, \quad \text { with } x=\frac{b}{a}
\end{equation} or \begin{equation}
\circ(a+b)^{n}=b^{n}(1+y)^{n}, \quad \text { with } y=\frac{a}{b}
\end{equation}

\begin{equation}
(a+x)^{n}=\sum_{k=0}^{n}\left(\begin{array}{c}{n} \\ {k}\end{array}\right) x^{k} a^{n-k},\left(\begin{array}{c}{n} \\ {k}\end{array}\right)=\frac{n !}{k !(n-k) !}
\end{equation}

\begin{equation}
(1+r)^{-1}=1-r+r^{2}-r^{3}+\cdots
\end{equation}

\begin{equation}
a\left(\frac{1-r^{n}}{1-r}\right)=a+a r+a r^{2}+a r^{3}+\cdots
\end{equation}

\begin{equation}
\sum y^{i}=1+y+y^{2}+\cdots \cdots=\frac{1}{1-y},(y<1)
\end{equation}

\begin{equation}
\sum i y^{i}=y\left(1+2 y+3 y^{2}+\cdots \cdots\right)=\frac{y}{(1-y)^{2}},(y<1)
\end{equation}

Trigonometric Formula

to continue...

The chemical structure below is generated on the fly from 'Chemaxon Smiles' format

[chem]CC1=C(C)C(C)=C(C)C(C)=C1C |c:1,9,t:5|[/chem]

Steps to create  the chemical structures

1. Visit: https://marvinjs-demo.chemaxon.com/latest/demo.html 

Draw the molecule/reaction of your choice using drawing tools.

2. After completing your drawing, click save button (marked as 1 in figure below) and a new window will pop-up to select ChemAxon Smiles format from the dopdownbox (marked as 2 in figure below) 

 

3. Select the content and use Ctrl+c to copy the text. This defines the structure of your molecule

4. You can now insert this string inside course contents/activities/questions/resources anywhere using the following special html format

Note the copied Smiles format is enclosed between two 'chem' tags. Note the beginning and end tag formats. You should not change this markup format. 

When the document is displayed the image will appear as

[chem]CC1=CNC=C1 |c:4,t:1,lp:3:1|[/chem]

The image produced with a default dimension of 200px by 200px

5. You can change the size of the image by adding them after the string like below 

please note the '@w' for width description and '@h' for height description separator and you should not use any other separator. The first number represents width and second represents height of the image

The following image is scaled to a custom size of 100x100 [chem]CC1=C(C)C(C)=C(C)C(C)=C1C |c:1,9,t:5|@d100@d100[/chem]

Please note that image appears only after saving the document.

-----------------------------------------------------

How to display chemical reactions

1. Go to https://marvinjs-demo.chemaxon.com/latest/demo.html 

2. Draw your reactions using the tools

3. Press save button and choose 'ChemAxon Marvin Document' from the format dropdown box appearing in the pop-up window

4. Copy the whole content (the text will be in xml format)

5. Insert the content copied, step 4 above, between special  html tag

6. You should see something like this (reaction is scaled to 400x200)

[chem]<cml xmlns="http://www.chemaxon.com" version="ChemAxon file format v20.20.0, generated by vunknown" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" xsi:schemaLocation="http://www.chemaxon.com http://www.chemaxon.com/marvin/schema/mrvSchema_20_20_0.xsd"><MDocument><MChemicalStruct><reaction><arrow type="DEFAULT" x1="-1.7498987658511798" y1="-1.0299797913210234" x2="2.041767900815487" y2="-1.0299797913210234"/><reactantList><molecule molID="m1"><atomArray><atom id="a1" elementType="C" x2="-4.583325432517098" y2="0.42678019635822917"/><atom id="a2" elementType="C" x2="-5.916872088515392" y2="-0.3432197974817699"/><atom id="a3" elementType="C" x2="-5.916872088515392" y2="-1.8834064518269438"/><atom id="a4" elementType="C" x2="-4.583325432517098" y2="-2.6534064456669437"/><atom id="a5" elementType="C" x2="-3.2495921098536336" y2="-1.8834064518269438"/><atom id="a6" elementType="C" x2="-3.2495921098536336" y2="-0.3432197974817699"/><atom id="a7" elementType="C" x2="-4.583372101237295" y2="1.9667801956510962"/><atom id="a8" elementType="O" x2="-5.917074556813034" y2="2.7367397790002768" lonePair="2"/></atomArray><bondArray><bond id="b1" atomRefs2="a1 a2" order="2"/><bond id="b2" atomRefs2="a2 a3" order="1"/><bond id="b3" atomRefs2="a3 a4" order="2"/><bond id="b4" atomRefs2="a4 a5" order="1"/><bond id="b5" atomRefs2="a5 a6" order="2"/><bond id="b6" atomRefs2="a6 a1" order="1"/><bond id="b7" atomRefs2="a1 a7" order="1"/><bond id="b8" atomRefs2="a7 a8" order="2"/></bondArray></molecule></reactantList><agentList/><productList><molecule molID="m2"><atomArray><atom id="a1" elementType="C" x2="4.583341234149569" y2="0.34344686302489613"/><atom id="a2" elementType="C" x2="3.249794578151274" y2="-0.42655313081510293"/><atom id="a3" elementType="C" x2="3.249794578151274" y2="-1.9667397851602768"/><atom id="a4" elementType="C" x2="4.583341234149569" y2="-2.7367397790002768"/><atom id="a5" elementType="C" x2="5.917074556813033" y2="-1.9667397851602768"/><atom id="a6" elementType="C" x2="5.917074556813033" y2="-0.42655313081510293"/><atom id="a7" elementType="C" x2="4.583294565429372" y2="1.8834468623177631"/><atom id="a8" elementType="O" x2="3.2495921098536336" y2="2.6534064456669437" lonePair="2"/><atom id="a9" elementType="O" x2="5.916950352284914" y2="2.6534872782614496" lonePair="2"/></atomArray><bondArray><bond id="b1" atomRefs2="a1 a2" order="2"/><bond id="b2" atomRefs2="a2 a3" order="1"/><bond id="b3" atomRefs2="a3 a4" order="2"/><bond id="b4" atomRefs2="a4 a5" order="1"/><bond id="b5" atomRefs2="a5 a6" order="2"/><bond id="b6" atomRefs2="a6 a1" order="1"/><bond id="b7" atomRefs2="a1 a7" order="1"/><bond id="b8" atomRefs2="a7 a8" order="2"/><bond id="b9" atomRefs2="a7 a9" order="1"/></bondArray></molecule></productList></reaction></MChemicalStruct><MTextBox toption="NOROT" fontScale="14.0" halign="LEFT" valign="TOP" autoSize="true" id="o1"><Field name="text">[O]</Field><MPoint x="-0.9165654325178458" y="0.1366868753456436"/><MPoint x="-0.9165654325178458" y="0.1366868753456436"/><MPoint x="-0.9165654325178458" y="0.1366868753456436"/><MPoint x="-0.9165654325178458" y="0.1366868753456436"/></MTextBox></MDocument></cml>@d400@d200[/chem]

7. The default size is 200x200. You can also change the size by using '@d  and '@d separator

---------------------------

An example of a big molecule (using Chemaxon Smiles Format)

[chem]CC(C)CC1NC(=O)C(CCC(N)=O)N(C)C(=O)C(NC(=O)C(CS([O-])(=O)=O)NC(=O)C(CCC[NH+]=C(N)N)NC(=O)C(Cc2c[nH]c3ccccc23)NC(=O)C(NC(=O)C(NC(=O)C2CCCN2C(=O)C(Cc2ccccc2)NC(=O)C(C)NC=O)C(C)(C)C)C(C)(C)C)C(C)C(=O)CN(C)C(=O)C(NC(=O)C(CC(N)=O)NC1=O)C(C)O |lp:5:1,7:2,12:1,13:2,14:1,17:2,19:1,21:2,25:3,26:2,27:2,28:1,30:2,37:1,38:1,39:1,41:2,53:1,55:2,57:1,59:2,61:1,63:2,68:1,70:2,79:1,81:2,84:1,86:2,98:2,100:1,103:2,105:1,107:2,111:1,112:2,113:1,115:2,118:2|@d500@d500[/chem]

Another one with stereo-chemistry descriptions

[chem]CC(C)(O1)C[C@@H](O)[C@@]1(O2)[C@@H](C)[C@@H]3CC=C4[C@]3(C2)C(=O)C[C@H]5[C@H]4CC[C@@H](C6)[C@]5(C)Cc(n7)c6nc(C[C@@]89(C))c7C[C@@H]8CC[C@@H]%10[C@@H]9C[C@@H](O)[C@@]%11(C)C%10=C[C@H](O%12)[C@]%11(O)[C@H](C)[C@]%12(O%13)[C@H](O)C[C@@]%13(C)CO@d400@d400[/chem]

--------------------------

A reaction display using Chemaxon smiles format

[chem]CC([O-])[N+](C)C>>CNC.CC=O |f:1.2,lp:2:3,7:1,11:2,^1:3|@d400@d200[/chem]

-----------

Curved arrow display (mrv format)

[chem]<cml xmlns="http://www.chemaxon.com" version="ChemAxon file format v18.11.0, generated by vunknown" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" xsi:schemaLocation="http://www.chemaxon.com http://www.chemaxon.com/marvin/schema/mrvSchema_18_11_0.xsd"><MDocument><MChemicalStruct><reaction><arrow type="DEFAULT" y2="-0.06856343529221298" x2="-1.971515409002313" y1="-0.06856343529221298" x1="-5.997486710203567"/><reactantList><molecule molID="m1"><atomArray><atom id="a1" y2="2.000518682742053" x2="-9.814512455161372" elementType="C"/><atom id="a2" y2="0.666839560914018" x2="-9.044512455161371" elementType="C"/><atom id="a3" y2="0.6668395609140191" x2="-7.504512455161372" elementType="O" lonePair="3" formalCharge="-1"/><atom id="a4" y2="-0.6668395609140176" x2="-9.81451245516137" elementType="N" formalCharge="1" radical="monovalent"/><atom id="a5" y2="-2.0005186827420535" x2="-9.044512455161371" elementType="C"/><atom id="a6" y2="-0.6668395609140176" x2="-11.35451245516137" elementType="C"/></atomArray><bondArray><bond id="b1" order="1" atomRefs2="a1 a2"/><bond id="b2" order="1" atomRefs2="a2 a3"/><bond id="b3" order="1" atomRefs2="a2 a4"/><bond id="b4" order="1" atomRefs2="a4 a5"/><bond id="b5" order="1" atomRefs2="a4 a6"/></bondArray></molecule></reactantList><agentList/><productList><molecule molID="m2"><atomArray><atom id="a1" y2="0.16948131725794724" x2="1.2784750896772614" elementType="C"/><atom id="a2" y2="-0.6005186827420532" x2="2.612154211505297" elementType="C"/><atom id="a3" y2="0.16948131725794724" x2="3.9458333333333324" elementType="O" lonePair="1" formalCharge="1" mrvValence="2"/></atomArray><bondArray><bond id="b1" order="1" atomRefs2="a1 a2"/><bond id="b2" order="2" atomRefs2="a2 a3"/></bondArray></molecule><molecule molID="m3"><atomArray><atom id="a1" y2="0.5900000000000005" x2="8.687154211505298" elementType="C"/><atom id="a2" y2="-0.1799999999999995" x2="10.020833333333332" elementType="N" lonePair="1"/><atom id="a3" y2="0.590000000000001" x2="11.354512455161367" elementType="C"/></atomArray><bondArray><bond id="b1" order="1" atomRefs2="a1 a2"/><bond id="b2" order="1" atomRefs2="a2 a3"/></bondArray></molecule></productList></reaction></MChemicalStruct><MElectronContainer id="o1" radical="0" occupation="0 0"><MElectron difLoc="0.0 0.0 0.0" atomRefs="m1.a3"/><MElectron difLoc="0.0 0.0 0.0" atomRefs="m1.a3"/></MElectronContainer><MElectronContainer id="o2" radical="0" occupation="1 1"><MElectron difLoc="0.0 0.0 0.0" atomRefs="m1.a3"/><MElectron difLoc="0.0 0.0 0.0" atomRefs="m1.a3"/></MElectronContainer><MElectronContainer id="o3" radical="0" occupation="0 0"><MElectron difLoc="0.0 0.0 0.0" atomRefs="m1.a3"/><MElectron difLoc="0.0 0.0 0.0" atomRefs="m1.a3"/></MElectronContainer><MElectronContainer id="o4" radical="1" occupation="0"><MElectron difLoc="0.0 0.0 0.0" atomRefs="m1.a4"/></MElectronContainer><MElectronContainer id="o5" radical="0" occupation="0 0"><MElectron difLoc="0.0 0.0 0.0" atomRefs="m2.a3"/><MElectron difLoc="0.0 0.0 0.0" atomRefs="m2.a3"/></MElectronContainer><MElectronContainer id="o6" radical="0" occupation="0 0"><MElectron difLoc="0.0 0.0 0.0" atomRefs="m3.a2"/><MElectron difLoc="0.0 0.0 0.0" atomRefs="m3.a2"/></MElectronContainer><MEFlow id="o7" arcAngle="-216.66065987595"><MAtomSetPoint atomRefs="m1.a2 m1.a4"/><MAtomSetPoint atomRefs="m1.a4"/></MEFlow><MEFlow id="o8" arcAngle="-173.73342937584172" baseElectronIndexInContainer="-1" baseElectronContainerIndex="1"><MEFlowBasePoint atomRef="m1.a3"/><MAtomSetPoint atomRefs="m1.a2"/></MEFlow></MDocument></cml>@d400@d200[/chem]