Legrange's Mean Value Theorem

Let f : [a, b] → R be a continuous function on the closed interval [a, b], and differentiable on the open interval (a, b), where a < b. Then there exists some c in (a, b) such that

The mean value theorem is a generalization of Rolle's theorem, which assumes f(a) = f(b), so that the right-hand side above is zero.
The mean value theorem is still valid in a slightly more general setting. One only needs to assume that f : [a, b] → R is continuous on [a, b], and that for every x in (a, b) the limit

exists as a finite number or equals +∞ or −∞. If finite, that limit equals f′(x). An example where this version of the theorem applies is given by the real-valued cube root function mapping x to x^1/3, whose derivative tends to infinity at the origin.
Note that the theorem, as stated, is false if a differentiable function is complex-valued instead of real-valued. For example, define f(x) = e^ix for all real x. Then

f(2π) − f(0) = 0 = 0(2π − 0)

while |f′(x)| = 1.

Torricelli's (Roger's) law

Torricelli's (Roger's) law, also known as Torricelli's theorem, is a theorem in fluid dynamics relating the speed of fluid flowing out of an opening to the height of fluid above the opening.

Torricelli's law states that the speed of efflux, v, of a fluid through a sharp-edged hole at the bottom of a tank filled to a depth h is the same as the speed that a body (in this case a drop of water) would acquire in falling freely from a height h, i.e. , where g is the acceleration due to gravity (9.8 N/kg). This last expression comes from equating the kinetic energy gained, , with the potential energy lost, mgh , and solving for v.

Definition of the Day -Heat Of Combustion

The heat of combustion () is the energy released as heat when a compound undergoes complete combustion with oxygen under standard conditions. The chemical reaction is typically a hydrocarbon reacting with oxygen to form carbon dioxide, water and heat. It may be expressed with the quantities:

• energy/mole of fuel (kJ/mol)
• energy/mass of fuel
• energy/volume of fuel

Organic Diagrams and Projections- Newman projections

Wedge-dash diagrams
Usually drawn with two bonds in the plane of the page, one infront, and one behind to give the molecule perspective. When drawing wedge-dash it is a good idea to visualise the tetrahedral arrangement of the groups and try to make the diagram "fit" this. As a suggestion, they seem to be most effective when the "similar" pairs of bonds (2-in-plane, 2-out-of-plane) are next to each other, see below:

Sawhorse
Sawhorse diagrams are similar to wedge-dash diagrams, but without trying to use "shading" to denote the perspective. The representation to the right of propane has been drawn so that we are looking at the molecule which is below us and to our left.
Newman Projections
These projections are drawn by looking directly along a particular bond in the system (here a C-C bond) and arranging the substituents symmetrically around the atoms at each end of that bond. The protocol requires that the atoms within the central bond are defined as shown below:

In order to draw a Newman projection from a wedge-dash diagram, it is useful to imagine putting your "eye" in line with the central bond in order to look along it.
Let's work through an example, consider drawing a Newman projection by looking at the following wedge-dash diagram of propane from the left hand side.

• First draw the dot and circle to represent the front and back C respectively
• Since the front carbon atom has an H atom in the plane of the page pointing up we can add that first
• The back carbon atom has an H atom in the plane of the page pointing down
• Now add the other bonds to each C so that it is symmetrical
• The groups / bonds (blue) that were forward of the plane of the page in the original wedge-dash diagram are now to our right
• Those behind (green) the plane are now to our left

• Now you try the same thing, but looking from the right to generate the other Newman projection.

Drawing Cyclohexanes
Drawing cyclohexane so that it looks like a chair can be the key to appreciating the axial and equatorial positions. If you are unable to draw good looking structures that clearly show axial and equatorial positions, then your instructor is probably going to assume that you don't know.
By not mastering the trick of drawing cyclohexanes the only person that really suffers is you the student. You deprive yourself of the knowledge and the chance to appreciate it and what it means. Believe me, it will be needed later.
The first step is drawing the chair itself.  Although the chair "looks better" when slightly angled, it maybe easier to "learn" to draw it with the middle portion horizontal.
Cycloalkanes

• Cycloalkanes just means "cyclic alkanes" - ring systems formed of only C-C and C-H bonds.
• Such structures are commonly encountered in natural compounds such as steroids, with cyclopentanes and cyclohexanes being the most common.
• Other than cyclopropane (which must be planar), cycloalkanes are "puckered" to relieve some of the ring strain by lowering angle and torsional strains.
• Ring strain : cyclopropane > cyclobutane > cyclopentane > cyclohexane

The structures of some of the smaller cycloalkanes are shown below with the planar structures for contrast. In each case, manipulate the CHIME images to look for the deviation from planarity and the effect this has on the eclipsing interactions of adjacent H atoms and C-C bonds. In order to be able to compare the strain in each member of the cycloalkane series, the heat of combustion per methylene (i.e. -CH₂-) is also given. The smaller this number is the less ring strain there is.
 

C3H6 CYCLOPROPANE ΔHc / C7 kJ/mol (-166.6 kcal/mol)

 

C4H8 CYCLOBUTANE ΔHc / CH2 = -681 kJ/mol (-162.7 kcal/mol)

 

C5H10 CYCLOPENTANE ΔHc / CH2 = -658 kJ/mol (-157.3 kcal/mol)

Magnetism

Magnetism is a force of attraction or replusion that acts at a distance. It is due to a magnetic field, which is caused by moving electrically charged particles or is inherent in magnetic objects such as a magnet.
A magnet is an object that exhibits a strong magnetic field and will attract materials like iron to it. Magnets have two poles, called the north (N) and south (S) poles. Two magnets will be attacted by their opposite poles, and each will repel the like pole of the other magnet. Magnetism has many uses in modern life.
Questions you may have include:

• What is a magnetic field?
• What is a magnetic force?
• What is the relationship between magnetism and electricity?

This lesson will answer those questions.

Useful tool: Metric-English Conversion

Magnetic field

A magnetic field consists of imaginary lines of flux coming from moving or spinning electrically charged particles. Examples include the spin of a proton and the motion of electrons through a wire in an electric circuit.
What a magnetic field actually consists of is somewhat of a mystery, but we do know it is a special property of space.

Magnetic field or lines of flux of a moving charged particle

Names of poles

The lines of magnetic flux flow from one end of the object to the other. By convention, we call one end of a magnetic object the N or North-seeking pole and the other the S or South-seeking pole, as related to the Earth's North and South magnetic poles. The magnetic flux is defined as moving from N to S.

Magnets

Although individual particles such as electrons can have magnetic fields, larger objects such as a piece of iron can also have a magnetic field, as a sum of the fields of its particles. If a larger object exhibits a sufficiently great magnetic field, it is called a magnet.

Magnetic force

The magnetic field of an object can create a magnetic force on other objects with magnetic fields. That force is what we call magnetism.
When a magnetic field is applied to a moving electric charge, such as a moving proton or the electrical current in a wire, the force on the charge is called a Lorentz force.

Attraction

When two magnets or magnetic objects are close to each other, there is a force that attracts the poles together.

Force attracts N to S

Magnets also strongly attract ferromagnetic materials such as iron, nickel and cobalt.

Repulsion

When two magnetic objects have like poles facing each other, the magnetic force pushes them apart.

Force pushes magnetic objects apart

Magnets can also weakly repel diamagnetic materials.

Magnetic and electric fields

The magnetic and electric fields are both similar and different. They are also inter-related.

Electric charges and magnetism similar

Just as the positive (+) and negative (−) electrical charges attract each other, the N and S poles of a magnet attract each other.
In electricity like charges repel, and in magnetism like poles repel.

Electric charges and magnetism different

The magnetic field is a dipole field. That means that every magnet must have two poles.
On the other hand, a positive (+) or negative (−) electrical charge can stand alone. Electrical charges are called monopoles, since they can exist without the opposite charge.

Summary

Magnetism is a force that acts at a distance and is caused by a magnetic field. The magnetic force strongly attracts an opposite pole of another magnet and repels a like pole. The magnetic field is both similar and different than an electric field.

Surface Tension

Surface tension is a contractive tendency of the surface of a liquid that allows it to resist an external force. It is revealed, for example, in the floating of some objects on the surface of water, even though they are denser than water, and in the ability of some insects (e.g. water striders) to run on the water surface. This property is caused by cohesion of similar molecules, and is responsible for many of the behaviors of liquids.
Surface tension has the dimension of force per unit length, or of energy per unit area. The two are equivalent—but when referring to energy per unit of area, people use the term surface energy—which is a more general term in the sense that it applies also to solids and not just liquids.
In materials science, surface tension is used for either surface stress or surface free energy.

Causes

Diagram of the forces on molecules of a liquid

Surface tension prevents the paper clip from submerging.

The cohesive forces among liquid molecules are responsible for the phenomenon of surface tension. In the bulk of the liquid, each molecule is pulled equally in every direction by neighboring liquid molecules, resulting in a net force of zero. The molecules at the surface do not have other molecules on all sides of them and therefore are pulled inwards. This creates some internal pressure and forces liquid surfaces to contract to the minimal area.
Surface tension is responsible for the shape of liquid droplets. Although easily deformed, droplets of water tend to be pulled into a spherical shape by the cohesive forces of the surface layer. In the absence of other forces, including gravity, drops of virtually all liquids would be perfectly spherical. The spherical shape minimizes the necessary "wall tension" of the surface layer according to Laplace's law.
Another way to view surface tension is in terms of energy. A molecule in contact with a neighbor is in a lower state of energy than if it were alone (not in contact with a neighbor). The interior molecules have as many neighbors as they can possibly have, but the boundary molecules are missing neighbors (compared to interior molecules) and therefore have a higher energy. For the liquid to minimize its energy state, the number of higher energy boundary molecules must be minimized. The minimized quantity of boundary molecules results in a minimized surface area.
As a result of surface area minimization, a surface will assume the smoothest shape it can (mathematical proof that "smooth" shapes minimize surface area relies on use of the Euler–Lagrange equation). Since any curvature in the surface shape results in greater area, a higher energy will also result. Consequently the surface will push back against any curvature in much the same way as a ball pushed uphill will push back to minimize its gravitational potential energy.

Effects of surface tension

Water

Several effects of surface tension can be seen with ordinary water:
A. Beading of rain water on a waxy surface, such as a leaf. Water adheres weakly to wax and strongly to itself, so water clusters into drops. Surface tension gives them their near-spherical shape, because a sphere has the smallest possible surface area to volume ratio.
B. Formation of drops occurs when a mass of liquid is stretched. The animation shows water adhering to the faucet gaining mass until it is stretched to a point where the surface tension can no longer bind it to the faucet. It then separates and surface tension forms the drop into a sphere. If a stream of water were running from the faucet, the stream would break up into drops during its fall. Gravity stretches the stream, then surface tension pinches it into spheres.
C. Flotation of objects denser than water occurs when the object is nonwettable and its weight is small enough to be borne by the forces arising from surface tension. For example, water striders use surface tension to walk on the surface of a pond. The surface of the water behaves like an elastic film: the insect's feet cause indentations in the water's surface, increasing its surface area.
D. Separation of oil and water (in this case, water and liquid wax) is caused by a tension in the surface between dissimilar liquids. This type of surface tension is called "interface tension", but its physics are the same.
E. Tears of wine is the formation of drops and rivulets on the side of a glass containing an alcoholic beverage. Its cause is a complex interaction between the differing surface tensions of water and ethanol; it is induced by a combination of surface tension modification of water by ethanol together with ethanol evaporating faster than water.

• 

  A. Water beading on a leaf
  
• 

  B. Water dripping from a tap
  
• 

  C. Water striders stay atop the liquid because of surface tension
  
• 

  D. Lava lamp with interaction between dissimilar liquids; water and liquid wax
  
• 

  E. Photo showing the "tears of wine" phenomenon.
  

Surfactants

Surface tension is visible in other common phenomena, especially when surfactants are used to decrease it:

• Soap bubbles have very large surface areas with very little mass. Bubbles in pure water are unstable. The addition of surfactants, however, can have a stabilizing effect on the bubbles (see Marangoni effect). Notice that surfactants actually reduce the surface tension of water by a factor of three or more.

• Emulsions are a type of solution in which surface tension plays a role. Tiny fragments of oil suspended in pure water will spontaneously assemble themselves into much larger masses. But the presence of a surfactant provides a decrease in surface tension, which permits stability of minute droplets of oil in the bulk of water (or vice versa).

Basic physics

Two definitions

Diagram shows, in cross-section, a needle floating on the surface of water. Its weight, F_w, depresses the surface, and is balanced by the surface tension forces on either side, F_s, which are each parallel to the water's surface at the points where it contacts the needle. Notice that the horizontal components of the two F_s arrows point in opposite directions, so they cancel each other, but the vertical components point in the same direction and therefore add up to balance F_w.

Surface tension, represented by the symbol γ is defined as the force along a line of unit length, where the force is parallel to the surface but perpendicular to the line. One way to picture this is to imagine a flat soap film bounded on one side by a taut thread of length, L. The thread will be pulled toward the interior of the film by a force equal to 2L (the factor of 2 is because the soap film has two sides, hence two surfaces). Surface tension is therefore measured in forces per unit length. Its SI unit is newton per meter but the cgs unit of dyne per cm is also used. One dyn/cm corresponds to 0.001 N/m.
An equivalent definition, one that is useful in thermodynamics, is work done per unit area. As such, in order to increase the surface area of a mass of liquid by an amount, δA, a quantity of work, δA, is needed.This work is stored as potential energy. Consequently surface tension can be also measured in SI system as joules per square meter and in the cgs system as ergs per cm². Since mechanical systems try to find a state of minimum potential energy, a free droplet of liquid naturally assumes a spherical shape, which has the minimum surface area for a given volume.
The equivalence of measurement of energy per unit area to force per unit length can be proven by dimensional analysis.

Surface curvature and pressure

Surface tension forces acting on a tiny (differential) patch of surface. δθ_x and δθ_y indicate the amount of bend over the dimensions of the patch. Balancing the tension forces with pressure leads to the Young–Laplace equation

If no force acts normal to a tensioned surface, the surface must remain flat. But if the pressure on one side of the surface differs from pressure on the other side, the pressure difference times surface area results in a normal force. In order for the surface tension forces to cancel the force due to pressure, the surface must be curved. The diagram shows how surface curvature of a tiny patch of surface leads to a net component of surface tension forces acting normal to the center of the patch. When all the forces are balanced, the resulting equation is known as the Young–Laplace equation:

where:

• Δp is the pressure difference.
• is surface tension.
• R_x and R_y are radii of curvature in each of the axes that are parallel to the surface.

The quantity in parentheses on the right hand side is in fact (twice) the mean curvature of the surface (depending on normalisation).
Solutions to this equation determine the shape of water drops, puddles, menisci, soap bubbles, and all other shapes determined by surface tension (such as the shape of the impressions that a water strider's feet make on the surface of a pond).
The table below shows how the internal pressure of a water droplet increases with decreasing radius. For not very small drops the effect is subtle, but the pressure difference becomes enormous when the drop sizes approach the molecular size. (In the limit of a single molecule the concept becomes meaningless.)

Δp for water drops of different radii at STP
Droplet radius	1 mm	0.1 mm	1 μm	10 nm
Δp (atm)	0.0014	0.0144	1.436	143.6

Liquid surface

Minimal surface

To find the shape of the minimal surface bounded by some arbitrary shaped frame using strictly mathematical means can be a daunting task. Yet by fashioning the frame out of wire and dipping it in soap-solution, a locally minimal surface will appear in the resulting soap-film within seconds
The reason for this is that the pressure difference across a fluid interface is proportional to the mean curvature, as seen in the Young-Laplace equation. For an open soap film, the pressure difference is zero, hence the mean curvature is zero, and minimal surfaces have the property of zero mean curvature.

Contact angles

Main article: Contact angle

The surface of any liquid is an interface between that liquid and some other medium The top surface of a pond, for example, is an interface between the pond water and the air. Surface tension, then, is not a property of the liquid alone, but a property of the liquid's interface with another medium. If a liquid is in a container, then besides the liquid/air interface at its top surface, there is also an interface between the liquid and the walls of the container. The surface tension between the liquid and air is usually different (greater than) its surface tension with the walls of a container. And where the two surfaces meet, their geometry must be such that all forces balance.

Forces at contact point shown for contact angle greater than 90° (left) and less than 90° (right)

Where the two surfaces meet, they form a contact angle, , which is the angle the tangent to the surface makes with the solid surface. The diagram to the right shows two examples. Tension forces are shown for the liquid-air interface, the liquid-solid interface, and the solid-air interface. The example on the left is where the difference between the liquid-solid and solid-air surface tension, , is less than the liquid-air surface tension, , but is nevertheless positive, that is

In the diagram, both the vertical and horizontal forces must cancel exactly at the contact point, known as equilibrium. The horizontal component of is canceled by the adhesive force, .

The more telling balance of forces, though, is in the vertical direction. The vertical component of must exactly cancel the force, .

Liquid	Solid	Contact angle
water	soda-lime glass lead glass fused quartz	0°
ethanol
diethyl ether
carbon tetrachloride
glycerol
acetic acid
water	paraffin wax	107°
	silver	90°
methyl iodide	soda-lime glass	29°
	lead glass	30°
	fused quartz	33°
mercury	soda-lime glass	140°
Some liquid-solid contact angles

Since the forces are in direct proportion to their respective surface tensions, we also have

where

• is the liquid-solid surface tension,
• is the liquid-air surface tension,
• is the solid-air surface tension,
• is the contact angle, where a concave meniscus has contact angle less than 90° and a convex meniscus has contact angle of greater than 90°.

This means that although the difference between the liquid-solid and solid-air surface tension, , is difficult to measure directly, it can be inferred from the liquid-air surface tension, , and the equilibrium contact angle, , which is a function of the easily measurable advancing and receding contact angles (see main article contact angle).
This same relationship exists in the diagram on the right. But in this case we see that because the contact angle is less than 90°, the liquid-solid/solid-air surface tension difference must be negative:

Special contact angles

Observe that in the special case of a water-silver interface where the contact angle is equal to 90°, the liquid-solid/solid-air surface tension difference is exactly zero.
Another special case is where the contact angle is exactly 180°. Water with specially prepared Teflon approaches this. Contact angle of 180° occurs when the liquid-solid surface tension is exactly equal to the liquid-air surface tension.