Saturday, 23 March 2013

The close relation of work and heat

Work and heat are both measured in joules, and are considered different ways of transferring energy. They are found experimentally to be easily intercovertable. Consider the pistons and cylinders below:


We can perform work on the left piston to compress it. Or heat the gas on the right to increase the pressure, pushing the piston, hence performing work. You could even use some of the work gained from the expanding gas to push current through a resistor, producing heat, and that heat can be used to expand a gas... and so on.

Work is the transfer of energy which makes use of ordered motion. The ordered direction of a falling piston introduces random kinetic motion into the gas molecules. So the piston is doing work which is converted to heating the gas.

Heat is the transfer of energy which makes use of random motion. The expansion of a gas in the cylinder will convert the random motion of gas particles into the ordered upwards motion of the piston. So heat from the gas is transformed into work.

The ordered and random motion does not have to be kinetic. An electric current is considered to move in an ordered motion, so using energy to drive a current would be considered doing work.

Friday, 22 March 2013

van der Waal equation of state example


You are given the temperature, pressure, and constants a and b of a gas. Estimate the molar volume:

We simply multiple both sides by (Vm - b)Vm2 to get:


Then collect the powers of Vm to obtain:


Then plug the numbers into software to get a solution to this cubic equation.

van der Waals equation of state derivation


This attempts to account for some of the assumptions of the perfect gas equation of state. It is a good example of scientific thinking about a mathematical model, "model building" in other words. a and b are empirical parameters, but they can also be estimated.

First we take the ideal gas equation and try to account for volume taken up by the molecules themselves, by subtracting nb from the volume, where b is a constant.

This approximates molecules as hard spheres. So each molecule has a sphere of exclusion around it of 2r, anything less and the spheres would penetrated eachother.


This gives a total excluded volume of (4/3)π(2r)3 per molecule, which is 8Vmolecule.

This number is then halved "to prevent overcounting" to give 4Vmolecule. I have never been able to understand why this step is done.

But the end result is that b is roughly 4VNA. Since many molecules are quite soft, this number is typically the upper limit of empirical measurements.

The term on the right is because attractive forces will reduce both the rate of collisions of the molecules with the side of the container, and reduce the speed of these collisions. These forces are found to act with a strength proportional to the square of the molar concentration (n / V) of the molecules.

Remember that this is overwhelming an empirical law. And the justification given for it is only vague. There are other more satisfactory ones which you may come across later.

Thursday, 21 March 2013

Real gases


A real gas approaches a perfect gas at 0 pressure. Attractive forces dominate at moderate pressures and repulsive forces at high pressure - since the repulsive forces have a shorter range.

Hence a real gas is expected to be more compressible at moderate pressures and less compressible at high pressures.

Compression factor

The compression factor is the ratio of the molar volume a gas compared to the molar volume of a perfect gas, at the same pressure and temperature:


From the equation you can see that Z of a perfect gas is 1. Real gases with a larger than perfect volume have   Z > 1, and real gases with a lower than perfect volume have Z < 1.

From the argument in the first section we can expect Z to approach 1 and 0 pressure, be below 1 at moderate pressures, and above 1 at high pressure:


We can see this is true for most gases - H2 being a commonly-shown exception.

Partial pressures

In a mixture of gases, perfect or real, the partial pressure of an individual gas defined as:

pi = Patial pressure
p = Total pressure
xi = mole fraction of the gas


ni = moles of individual gas
n = total moles of all gases in the mixture

Note the following:

- Each mole fraction is between 1 and 0
- All the mole fractions in a mixture add up to one
- Total pressure is the sum of each partial pressure

These can also been understood intuitively by plugging in numbers.

If the gas is perfect, then Dalton's law applies:
The pressure exerted by a mixture of gases is the sum of the pressures that each one
would exist if it occupied the container alone.

Wednesday, 20 March 2013

Measuring pressure

A device to measure pressure is called a manometer. A device which measures atmospheric pressure is called a barometer. The simplest design for these is a column of mercury in a beaker:


When the atmospheric pressure is increase, force is applied on the mercury, pushing it up the column. Mechanical equilibrium is reached when this force is balanced by both increased pressure inside the column, and weight of the mercury inside the column.

If the column contains a vacuum, then the increased atmospheric pressure is balanced purely by the weight of the mercury.

Pressure is force divided by area. The weight of the lifted mercury is mg, so the pressure on the base of the column is this weight divided by cross-sectional area. This has to match the increase in atmospheric pressure for the system to be at equilibrium:

Pressure = mg / A

Since mass is volume*density:


Notice that pressure is independent of the cross-sectional area of the cylinder.

A more advanced solution is to use a curved tube:



The change in pressure is directly proportional to the difference in height of the two columns. A different of 1 mm corresponds to 1 mm Hg, also known as one torr.

The reason mercury is used is because of its low vapor pressure. Using a liquid like water would mean having to account for more vapor pressure in the sealed part of the tube - but it can be done.

Friday, 8 March 2013

Ethylene glycol

One of the most common acetal formation reactions you see in organic synthesis is the use of ethylene glycol. It is widely used as automotive antifreeze. A sweet-tasting, colorless, odourless liquid, which will kill pets or humans if they drink enough of it.


The TsOH is para-Toluenesulfonic acid. It is as strong as sulphuric acid, but exists as a stable white solid which makes it convenient to use. It is also non-oxidizing. And it dissolves in organic solvents.

Mechanism:


Don't miss out the formation of the oxonium ion.

A  five-membered cyclic acetal is known as a dioxolane, as created in this example.

This forms readily even from ketones. 6-ring diols work too. I don't know why ethylene glycol is more common, presumably it is cheaper to make.

An example of the use of this protecting group is below:


The reaction works if we first protect the carbonyl. Since acetals are stable to a base (cyclic acetals moreso), the grignard reagent will only attack the Br. Once we have used the Grignard reagent to attack an electrophile, we can then hydrolyse the acetal off using acid and water.

Notice that water needs to be distilled off for the protecting group to be fully added. Thankfully, ethylene glycol has a boiling point of 197 C. If the reactants or solvent had a boiling point lower than water, then we could remove water using a Dean Stark Head with benzene.