Internal energy and the first law of thermodynamics

  In previous posts, we've covered how molecules move randomly, and they have kinetic energy with it. This is called the internal energy. In ideal gas, molecules have different kinetic energy due to different speed but sum of KE are constant, when the gas is kept at a constant temperature.

  In real gas, because molecules exerts force to each other, making potential energy at any rate, at any instant. Random motion also varies potential energy randomly. For the real gas, internal energy is given by sum of potential energies and the kinetic energies.
However, If two objects with different temperature impacts together, temperature will flow from higher point to lower point, not by any difference in internal energies.

First law of thermodynamics

  The increase in internal energy of a system is equal to
the sum of the heat added to the system and the work done on it

In mathematical form, ΔU = Q + W
Be careful with the sign,
+ΔU means an increase in internal energy.
+Q means that heat has been added to the system.

+W means the work is done on the system. 

Getting an equation for average kinetic energy of a molecule

  Molecule has 3 component, which is x, y and z value for velocity. We'll take a look at just for x value, which would be 'Vx' for this example

Let's say container has length(or width) of L meters. Time taken for molecules to travel from one side to another side would be L/Vx (Distance/velocity = time taken). From one side to another side and back would be 2L/Vx.

When this molecule strikes wall at velocity 'Vx', since it is assumed to be perfectly elastic, it would bounce off with velocity '-Vx'. Thus, every collision with a wall will end up change momentum by

Δpx = 2mVx

  Since this is done in time 2L/Vx

px = 2mVx/(2L/Vx) = 2mVx * Vx/2L = mVx²

∴ px = mVx²/L

There are N number of molecules, and pressure is force over area, L²

p = NmVx²/ V

  Applying this to both x,y and z.. sounds hard but it is actually easier than you might think. Let's say same molecule has velocity of c, and using Pythagoras' theorem to three dimensions would be c² = c₁² + c₂² + c₃² where c₁, c₂ and c₃ are x,y and z component. And since we are dealing with large # of molecules, average value for all three component would be pretty much same, thus <c₁²> = <c₂²> = <c₃²>, this gives idea that <c₁²> = ⅓<c²>

Now, equation might be

pV = ⅓Nm<c²>

Since average kinetic energy of a molecule is <Ek> = ½m<c²>   (KE = ½mv²)
pV = ⅓N *  2       *  ½m<c²>∴ pV = ⅔N<Ek>

From ideal gas equation, pV = nRT = NkT
This is important.. because now we can relate temperature of the gas to kinetic energy, as increase in pressure by temperature means rise in speed of molecules, thus we can relate kinetic energy with temperature. Thus
pV = ⅔N<Ek> = NkT

Making <Ek> subject
<Ek> = 3/2kT


Secondly, we can get average speed of the molecule with this equation.
We already had <Ek> = ½m<c²>
<Ek> = ½m<c²> = 3/2kT

Solving for <c²>
<c²> = 3kT/m

And
<c²>^½ = 3kT/m^½

Where <c²>^½ is root-mean-square speed or r.m.s speed (crms)

The kinetic theory an idear gas

  Brownian motion indicates that molecules are moving about at great speed. This molecules, contacting with walls, exerts their momentum. As this is continuously happening, it causes the wall to experience some pressure by the impacts of the molecules. Expressed in equations, some assumptions are needed

1. All molecules behave as identical, hard and perfectly elastic spheres
2. The volume of the molecules is negligible compared with the vessel
3. No force/attraction/repulsion between molecules
4. There are many molecules, moving randomly

Supposing that the moles has velocity of x,y and z value, which is parallel to the edge of the container, average kinetic energy of a molecule in gas is

<Ek> = ½m<c^2>=⅔kT                  k = Boltzmann constant = 1.38x10^-23 J/K

Click here to see how this is derived!

Since adding up velocity of molecules will be 0, (x,y,z sums up to 0 because they have both + and - value for both direction) we add the square number of components and mean it (Pythagoras theory) to get velocity of the molecule.
root-mean-square speed of a molecule is

Crms = (3kT/m)^½

Brownian motion

  Molecules are moving around, rapidly, and randomly all the time, and this is called Brownian motion. It was named after botanist Robert Brown (1773-1858), who discovered random-jerky motion by observing pollen grains suspended in water. Even though water was completely still, pollen grains were still moving. However, he didn't found out why, assuming that it was evidence of life. It was Einstein that analysed theoretical Brownian motion.
  

  In liquid/gas state, because spaces between atoms are empty, motion of molecules are quite rapid, and random. But in solid, because atoms are compacted quite tightly and fixed at one point to form solid, this motion is restricted to vibration about their equilibrium positions. 

The Gas laws

Equation of state of the gas

In 17th~18th centuries, several scientists has done some experiments on gas and found that volume, pressure and temperature of gas is all related. This is represented with few laws; Boyle's law, Charles' law and Gay-Lussac's law(law of pressures). 

Boyle's Law
Named after Robert Boyle (1627-1691)

Robert Boyle found with his experiment that when pressure on the gas is lowed, volume gets bigger. And he also found this can be plotted as straight line that go through origin. This is simplified as 

The volume of the gas is inversely proportional to its pressure
when the temperature of the body is constant.

Mathematically written as V ∝ 1/p or pV = Constant 
Therefore, for same sample of gas

p1V1 = p2V2

Charles' Law
Named after Jacques Charles (1746-1823)

Charles' Law indicates how temperature affects on the volume of a gas. Because gas liquefies when temperature is reduced, experiments couldn't obtain any results below liquefaction temperature. However, if we plot the graph and project it backwards, it was found that it intersects with temperature axis at about -273℃, aka absolute zero (0ºK).
When Celsius temperature are converted to thermodynamic temperatures, it can be said as 

With same mass of gas at constant pressure,
the volume of the gas is directly proportional to its thermodynamic temperature

Mathematically written as V ∝ T or V/T = Constant 

Therefore, for same sample of gas

V1 / T1 = V2 / T2

Gay-Lussac's Law / law of pressures
Named after Joseph Gay-Lussac (1778-1850)

With same mass of the gas at constant volume,
The pressure of the gas is directly proportional to its thermodynamic temperature.

Mathematically written as p ∝ T or pT = Constant 

Therefore, for same sample of gas

p1 / T1 = p2 / T2

Ideal Gas function

Combining these three laws, we can get

pV ∝ T

But as all of these laws is related to mass of the gas (m), we can derive it into 

pV ∝ mT

Mass of the gas can be find with number of moles of the gas

Moles(㏖) are defined as amount of atoms there are in 0.012kg of carbon-12. This is related with Avogardro Constant, Which is 6.02x10^23/㏖(meaning number of atoms per one mol) As atomic mass unit is 1.66x10^-27㎏, Multiplying these two, we can get approximately 1g (defined as relative molecular mass[1/12 of one more of carbon-12]).

for TL;DR, this is shorten to (where n is number of moles of the gas)

pV ∝ nT

This can be represented with another constant, R 

pV = nRT

or

pV = NkT

Where N is number of molecules (NOT moles!)
and k is called Boltzmann constant. R and k is related with Avogardro constant  R = kNA

However, In real life, these laws doesn't really work because its not really accurate with high air pressure. These is true only if the pressure of the sample is low enough, and gas is not under liquefaction temperature. Still, they can be define as an ideal gas.

An ideal gas, is one which obeys the equation of state
pV = nRT
at all pressures, volumes and temperature. 

List of variables/constants and units in this page

V = Volume = ㎥

p = Pressure = ㎩

T = Temperature = ºK
m = Mass = ㎏

n = Number of moles = ㏖

NA = Avogadro Contonstant = 6.02x10^23/㏖
Ar = Relative atomic mass = mass of 1/12 an atom Carbon-12
Mr = Relative molecular mass = mass of 1/12 of a mol of Carbon 12 (1g) (Ar*NA)
u = atomic mass unit = 1.66x10^-27㎏

R = molar gas constant/universal gas constant = 8.3J/K
k = Boltzmann constant = 1.38x10^-23J/K

Thermometers

Thermometers

  Thermometers uses thermodynamic properties of materials to measure temperatures as mentioned in previous posts. Different types of thermometers has different purposes to measure temp at different situations, such as measuring extremely hot substance or rapidly varying temperature. For Several important properties of thermometers might be;

• Accuracy
• Sensitivity
• range of temperature
• time it takes to give you reading
• How much those thermometer absorbs heat

  For some applications, measuring temperature of small body (such as little amount of liquid) might not be accurate if we don't use thermometers that doesn't need to touch the body(such as laser thermometer gun). When we use thermometers that needs to touch the body (liquid-in-glass thermometers for example), thermometers might end up absorbing some heat from body, thus changing temperature of the body that drops accuracy.  

  Some thermometers uses varying resistor called 'thermistor'(which I'll talk about it later in depth  with other electrical sensors) that gives resistance depending temperature, allowing thermometers to give readings to temperature. This allows thermistor thermometers to give readings for rapidly varying temperature, or small substance if thermistor uses suitable semiconducting materials. Creating small logic circuit allows to create system that can process things due to temperature. One of common use of this thermometer might be system in the computers/laptops that shutdowns computer if they gets too hot that can damage the circuit.

Thermodynamic temperature

When pressure of real gas is decreased, it act as an ideal gas. Ideal gas has relationship between pressure p, volume V, temperature T is

pV/T = constant

T is thermodynamic temperature of the body, and measured in Kelvin (ºK)

One kelvin is the fraction of 1/273.16 of the

thermodynamic temperature of the triple point of water. 

To avoid complications and confusion between empirical centigrade scale based on experiments with real gases and constant-volume gas thermometer scale, a new scale, the Celsius scale was defined by international agreement which is 

Unit of Celsius scale, which is degree Celsius (℃)
has same scale as thermodynamic scale, which is Kelvin(ºK).

Thus, Temperature θ ℃= T/K - 273.15 or T/K = θ ℃ + 273.15

To avoid confusion of absolute zero, which is 273.16, and number used in this equation, 273.15, and for various purposes, it is sufficient to work with the number 273