Category Archives: Physical Chemistry

Thermodynamics and spontaneity

The first law of thermodynamics is basically the law of conservation of energy as relevant to thermodynamic systems. The change in internal energy of a system, (ΔU) is equal to the sum of the heat and work (q + w) of the system.
The second law of thermodynamics describes whether or not a change is spontaneous, expressing it in terms of entropy.

Entropy (S), is the thermodynamic quantity that describes the disorder, (randomness) in a system. The entropy is related to the number of states a molecule has available to it. A molecule at high temperature has more vibrational states available than one at a lower temperature, and therefore has a higher entropy. A crystal locks molecules into a certain configuration, whereas molecules in a gas are free to move about and therefore have higher entropy.

The second law of thermodynamics states that the total entropy of a system and its surroundings always increases for a spontaneous process. Generally, we refer to this is the entropy of the universe, as the sum of the entropies of the system and surroundings must increase, however one may decrease and the process may still be spontaneous.

ΔSuniverse = ΔSsystem +ΔSsurroundings

We can restate this law so it refers only to the system, and as heat flows into or out of the system, the entropy goes with it. So, at a certain temperature, the entropy is associated with heat q;

ΔS > q/T for a spontaneous process

Therefore, for a spontaneous process at a certain temperature, the change in the entropy must be greater than the heat divided by the absolute temperature. For systems that are at equilibrium, the entropy is equal to the heat over temperature.

Phase changes:

The entropy of a phase change is derived from the equation above.

ΔS=ΔH/T

Where the ΔH is the heat of the phase change, and the temperature at which the phase change occurs.

 Spontaneity:

Looking at entropy and enthalpy, we can determine whether or not a process is spontaneous, and we introduce the concept of free energy (sometimes called Gibbs’ free energy, ΔG), which is equal to:

ΔG = ΔH – TΔS

For a spontaneous process, ΔG ≤ ΔH – TΔS, (ie, negative). We want the TΔS term to be larger than the ΔH term, indicating that even if a reaction is endothermic, if ΔS is larger, the reaction will still proceed.

 

Some links:

http://www.learnchem.net/tutorials/spont.shtml

http://secondlaw.oxy.edu/

http://www.chem1.com/acad/webtut/thermo/entropy.html

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Introduction to thermodynamics

Thermochemistry, the study of heat and energy and its effects on chemical reactions and physical transformations is one of the first branches of the chemical sciences and took off in the 1780s.

In chemistry, we look at how systems change when we change the conditions they are in. In terms of thermodynamics, we want to know what happens to the energy of a system under certain conditions.

Two types of energy we concern ourselves with in this area are work and heat. They are related to one another, and can be transformed from one to another. For example, heat energy released by the combustion of fuel can be used to turn a turbine, or rubbing sticks together will heat them up.

Work (w):

In physics, work is defined as

w= F x d

 where F is the force exerted and d is the distance moved.

In chemistry we change the equation slightly. We are no longer talking about something moving in one dimension, but a three dimensional system, we change Δd to ΔV and the force being exerted is the pressure of the gas. Therefore, work becomes:

w= -PΔV

It is negative because in chemistry we define work as that done by the system, not on it.

Heat (q):

Heat energy is related to the motion of the molecules – the hotter it is the faster they move, and jiggle about.

We measure changes in heat energy by looking at changes in temperature, and the two are related via the heat capacity of the substance. The heat capacity is the amount of energy required to raise one unit of the substance by one K (or °C). Molar heat capacity is measured in J/mol.K and specific heat capacity is J/g.K.

 

 q = CΔT

Internal energy (ΔU):

The total internal energy of the system is the sum of the potential energy and the kinetic energy. In this case, the potential energy is the work energy, and kinetic is related to the heat energy:

ΔU = w + q

Enthalpy (ΔH):

Enthalpy is a bit weird. It is the internal energy of a reaction plus any work it needs to do against the constant pressure of the atmosphere. My analogy is to imagine there is a magician who is going to make a rabbit appear:

In order to create the rabbit, (assuming he is actually magical and not just concealing one in his hat), he has to use a certain amount of energy, (internal energy of a rabbit). Next, he needs to create enough space for the rabbit to appear into, so he needs to exert some work to push back the atmosphere in the rabbit-shaped space (work). The enthalpy is the total energy expended by our magician:

 ΔH = ΔU +PΔV

Or

Energy expended by magician = internal energy of a rabbit + work to move the atmosphere

Meanwhile, the rabbit is left with existential dilemmas for which we have no equations

The real trick to understanding how these things fit together, is to look at what happens under certain circumstances:

Constant temperature:

Under constant temperature, ΔT = 0. That is fairly self-explanatory. If there is no change in temperature, then the internal energy of the system is constant, ΔU= 0, so q = w. In other words, any change in heat energy is exactly countered by work being done by/on the system so there is no net temperature change.

Constant volume:

If ΔV = 0, then there can be no work done, (w=0), so q = ΔU.

Constant pressure:

This is the most common condition for chemical reactions, as the external pressure (ie, the atmosphere) remains constant for anything done in an open container, (like, say, a beaker). At constant pressure, ΔH = q.

Some links:

http://www.ulb.ac.be/sma/testcenter/Test/problems/problems.html

http://www.shodor.org/unchem/advanced/thermo/

http://www.chem1.com/acad/webtext/energetics/

 

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Kinetic theory of gases

We can derive the ideal gas law by looking at the empirical laws of Charles, Boyle and Avogadro, but we can also find it theoretically using the kinetic theory of gases.

Hooke, an assistant of Boyle was the first to propose the idea that gas pressure is a result of collisions of randomly moving gas molecules/atoms with the sides of the container, (the explanation we use today). He didn’t advertise this idea though, as Newton had proposed an alternative, (and wrong) theory and Newton was a man that you simply didn’t disagree with. Hooke’s reputation isn’t all that great though – he may have been a brilliant man, (he developed the microscope, coined the term “cell” to describe… cells, developed Hooke’s law to describe the forces in a spring as well as contributions to astronomy, mechanics and the theory of gravitation. It was the latter that got him into trouble with Newton) but was apparently not a very nice one.

The kinetic theory is based on the idea that gas molecules/atoms are constantly, randomly moving and relies on the following postulates:

1. Gases are composed of molecules whose size is negligible, (compared to the distance between them).
2. Molecules move about randomly, but in straight lines and all at different speeds.
3. They don’t interact (attract or repulse) with each other.
4. When they do collide, the interaction is elastic (the total kinetic energy is maintained)
5. The average kinetic energy (KEav= 3/2RT) is proportional to the temperature.

To get the ideal gas equation, we remember that pressure is the result of a collision with the sides of the container and is therefore proportional to the frequency of the collisions and the average force exerted by the molecule during a collision.

P α frequency of collisions x force

The force depends on its momentum (its mass (m) x its velocity (u) – the bigger and faster it is, the more force it exerts. In other words, it hurts more to be hit with a fast bowling ball than a slow table tennis ball). The frequency of the collisions is dependant on the size of the container – a larger container will have fewer collisions, and on the speed of the molecules – the faster they are, the more often they will collide with the walls. Also, obviously, the more molecules you put in the container, the more often they will collide.

So… putting all of those things together:

P α (u x 1/V x N) x mu

If you put volume on the left:

PV α Nmu2

Kinetic energy depends on the mass and velocity of the molecules too (KE = 1/2mu2, which is also proportional to temperature).

PV α nT

(We can change N (number of molecules) to n (moles of molecules)) and when we put a proportionality constant in there we get the ideal gas equation!

PV = nRT.

Ta da!

The postulates for kinetic theory and the ideal gas equation are all well and good for ideal gases but they fall down for real gases. We know from experience that Boyle’s law is not valid had high pressure, (the gas will liquefy and the relationship between pressure and volume will no longer hold) and Charles’ law requires temperatures above the point where the gas will become liquid. These are true because in real life, gas particles will interact with one another, and are generally attracted to each other. They also have a real, measurable volume and cannot be compressed indefinitely. When considering real gases, we need to take these things into account and the ideal gas equation becomes the Van der Waals equation:

This version of the ideal gas equation has a correction factor (a) for the attractive forces between the molecules/atoms and (b) the real volume of the molecules/atoms.

 

Links:

http://www.science.uwaterloo.ca/~cchieh/cact/c120/gaskinetics.html

http://chemed.chem.wisc.edu/chempaths/GenChem-Textbook/Kinetic-Theory-of-Gases-Postulates-of-the-Kinetic-Theory-938.html

http://chemwiki.ucdavis.edu/Physical_Chemistry/Physical_Properties_of_Matter/Gases/Kinetic_Theory_of_Gases

 

 

 

 

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Gas laws

Discovery of the gas laws was one of the first steps chemistry took away from alchemy, largely thanks to Robert Boyle. These laws showed that there are simple relationships between the pressure, volume and amount of a gas and the combination of these empirical gas laws gives us the ideal gas law.

Boyle’s law:

Robert Boyle... and a very impressive wig

Robert Boyle was a 17th century alchemist, sometimes referred to as the “grandfather of chemistry”. He was involved in the foundation of the Royal Society, believed that all matter consisted of indivisible “corpuscles” and defined an element in the sense that we know it now. He also wrote some seriously nasty letters, (by 17th century standards) to Isaac Newton. We know him best for his work with gases, and he gave us Boyle’s Law. Using a manometer he showed that the volume of an enclosed gas decreases as the pressure increases.

PV=k or; the volume of a sample of gas at a given temperature varies inversely with the applied pressure.

Charles’ Law:

Jacques Charles - a ballooning science-pirate?

Jacques Charles was a bit awesome. Wikipedia describes him as an inventor, scientist, mathematician and balloonist, and he stole Charles’ law from a guy called Gay-Lussac. The law comes from hot-air-ballooning and describes how the volume of a gas increases linearly with temperature.

This is true up to a point – gases will liquefy before they reach zero volume, so Charles’ law does not hold for liquids.

 V=bT or; the volume occupied by any sample of gas at a constant pressure is directly proportional to the absolute temperature.

Avogardro’s law: 

Amedeo Avogadro, for whom 6.02x10^23 is named.

Avogadro (did you know that he didn’t actually calculate Avogadro’s number? It’s just named in his honour) after seeing what Gay-Lussac was doing, Avogadro showed that if you have equal volumes of gas at the same temperature and pressure, they have the same number of particles.

 V=an

 

The useful stuff comes when you combine the powers of Boyle, Charles and Avogadro, (and Gay-Lussac) and get the ideal gas law:

 

PV=nRT.  (this should be burned into your memory. Never forget this formula)

 

It shows that when gases behave ideally, (topic of a whole other post) the pressure, volume, temperature and molar amount can be calculated using the ideal gas formula. R is the ideal gas constant and is the constant of proportionality. It is 8.314 J.K-1.mol-1. Remember this, and don’t bother with the version that uses atmospheres.

 

A note about units: 

You will come across lots of units when dealing with gases, and it is useful to know what is equivalent to what.

 

1 mL = 1cm3

1 L = 1 dm3 (1 dm = 10 cm)

1000 L = 1 m3

 

1 atm = 1.01325 x 105 Pa = 101.325 kPa = 760 mmHg

 

You get a choice in these calculations – you can use m3, in which case you must use Pa or you can use L and use kPa. Decide now on one set of units to use from now on, and stick to it. If you chop and change, you will confuse yourself during an exam.

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Filed under Chem 1, Fundamentals, Physical Chemistry