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Published: Wednesday, 26 July 2023 15:02

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USA Schools / Colleges:

Schools:

In USA, there are public schools funded by State Governments, which provide high quality education. Unlike 3rd world countries where private schools are all the rage, here in USA you will see very few kids going to private school (mostly kids of wealthy families).

We have grades in USA schools. They are NOT called classes as in India. So, a student in matriculation (India 10th class) would be referred to being in "10th grade". Classes refer to different section (In India, it's section A, section B, etc) within that grade. So, when we say "what class are you?", it means within your grade, what section are you in? They don't have Sections as "Section A", etc here. This is because in middle and high schools, there are very few core courses. Beyond those core courses, you are free to take any course as per your liking. So, you change classes every hour and sit with totally new set of kids in almost each class. So, you don't really have a definite assigned "class" where you sit all day. Elementary schools is different as there are no electives there, so you do have an "assigned class". These classes are usually known by the name of the teacher who is teaching this class (i.e "Miss Elizabeth Class").

In public schools, education starts from Kindergarten (aka KG in Indian schools), goes to 1st grade (NOT called class as in India but grade), and then progresses to 12th grade (equivalent of 10+2 in India). Public schools are divided in 3 separate campuse to catr to kids of different ages. The youngest kids from Kindergarten - 5th grade go to Elementary school, from 6th - 8th grade go to Middle school and from 9th - 12th grade go to High School. The division is not very strict, and different school district may choose their own criteria (i.e middle school may go from 7th - 9th grade).

In section "USA Basic Facts", we saw that there are about 55M students in USA Schools. These are the age group composition:

• < 5 years: 6M kids in this age group go to some form of Preschool. They are not counted as students, as formal education starts from age 5+ thru Kindergarten class.
• 5-17 yrs: All 55M kids in this age group are attending some form of school (50M in public and 5M in private). 38M are in elementary/middle school, while 16M are in high school.
• 18-24 yrs: Out of 30M people in this age group, 1.5M are still enrolled in high school. This makes sense as Kindergarten starts from age 5, and adding 13 yrs of school education, takes it to 18 yrs of age. Kids who start b/w 5-6 yrs of age are the ones who are in High school even when they cross 18.

From above data, we may deduce that about 4M students graduate from High School every year (16M/4 years of high school = 4M).

Colleges:

In USA, there is no dearth of colleges to go to, as long as you can afford it. There are 2 year and 4 year colleges.

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Published: Tuesday, 18 July 2023 17:23

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Advanced Maths

This contains general topics in High School Maths that are not covered anywhere else. Calculus has it's own section.

Induction:

Induction is a vey common technique used to prove theorems that apply to natural numbers n. In absence of induction, such formula are really tedious to prove, but with Induction, they become a piece of cake. Some Maths Olympiad questions involve prove by Induction, so it's good to know this. Induction ONLY roves an existing formula, but can't find out the formula for a new question.

Wiki: https://en.wikipedia.org/wiki/Mathematical_induction

Proof by Induction: Given a Statement P(n) true for every natural number n (or for all natural numbers n >= N), we can use induction to prove it. This is done in 2 steps:

• First prove that the statement is true for n=0 or n=N (if P(n) is true for n>=N).
• Now assume P(n) is true for some n=k. Now, we try to prove that stmt holds for n=(k+1). We find P(n+1) by using P(n) and other maths transformations. We rearrange the terms so that this finally comes in same form as P(n+1). So, this proves that if P(n) holds, then P(n+1) will also hold. Since P(0) is already shown to be true, P(n) is true for all n. NOTE: We can't use formula for P(n+1) while finding P(n+!) since we haven't proved P(n+1) yet.

Sum of 1st N natural numbers: Prove by induction that sum of 1st n natural numbers is n(n+1)/2

This is the most common problem used to prove by induction. Here we have P(n)=n(n+1)/2.

We prove for n=1, i.e P(1)=1(1+1)/2=1 => True

Assume P(n)-n(n+1)/2. So P(k)=k(k+1)/2.

Sum of 1st (k+1) natural numbers = P(k)+k+1 = k(k+1)/2 + (k+1) = (k+1)(k+2)/2 = (k+1)((k+1)+1)/2. This is actually same as P(k+1) as P(k+1)=(k+1)(k+2)/2.

So, assuming Sum of 1st (k) natural numbers = P(k), then Sum of 1st (k+1) natural numbers turns out to be = P(k+1). So, if P(1) is true, then P(2) is true. If P(2) is true then P(3) is true and so on till infinity.

Sum of Squares of 1st N natural numbers: Prove by induction that sum of squares  of1st n natural numbers is n(n+1)(2n+1)/6

Here we have P(n)=n(n+1)(2n+1)/6.

We prove for n=1, i.e P(1)=1*2*3/6=1 => True

Assume P(n)-n(n+1)(2n+1)/6. So P(k)=k(k+1)(2k+1)/6.

Sum of squares of 1st (k+1) natural numbers = P(k)+(k+1)^2 = k(k+1)(2k+1)/6 + (k+1)^2 = (k+1)[k(2k+1)/6 + (k+1)] = (k+1)[2k^2+7k+6]/6 = (k+1)[(2k+3)(k+2)]/6. This is actually same as P(k+1) as P(k+1)=(k+1)(k+2)(2k+3)/6. Hence proved.

Induction technique above doesn't find out the formula, but instead just proves it. We can find out the formula for sum of squares by a neat technique. It's explained under "High school Algebra" section.

Similarly for sum of cubes of 1st n numbers can be proved by induction.

Complex Numbers:

First of all, complex numbers are not complex at all. Infact, problems involving complex numbers are usually simple to solve. There's always at least on question on Complex Numbers in US Maths olympiad, and it's relatively easy to solve. So, you should definitely master this.

Secondly, there's no real use of complex numbers in real life. They are a way to solve complicated equations in Physics, electronic, etc by transforming to complex plane. In this maths section, we'll just look at how to work with these.

Wiki => https://en.wikipedia.org/wiki/Complex_number

Imaginary number = √(-1) = i (English letter i)

i^n = ?

• i^1 = i
• i^2 = -1
• i^3 = i^2*i = -i
• i^4 = i^2*i^2 = 1
• i^n = +1, +i, -1, -i, depending on whether n/4 gives a remainder of 0,1,2,3

Complex number is any number with 2 parts: a real number (Re) added to an imaginary number (Im). Represented by Z. Z=a+b*i => Here a and b are both real, where a is the real part of Z and b is the imaginary part of Z. We usually plot complex numebrs on x,y ais, where x axis is the real part of Z, and Y axis is the Im part of Z. So, now we can plot coordinates of X not only in cartesian coordinates as (a,b), but also in polar coordinates (r,theta). Th polar form is much more helpful in solving complex arithmetic. Magnitude of Z (or r in polar coord) is defined as distance from origin r = √(a^2+b^2) and angle theta is tan^-1 (b/a). Magnitude of Z is also noted as |Z|

Addition/subtraction = Z1+Z2 = (a1+b1*i) + (a2+b2*i) = (a1+a2) + (b1+b2)*i  => (real parts add/sub and Im parts add/sub)

Multiplication: Z1*Z2 = (a1+b1*i) * (a2+b2*i) => Use algebraic multiplication to get (a1*a2 - b1*b2) + (a1*b2 + a2*b1)*i

Division: Z1/Z2 = (a1+b1*i) / (a2+b2*i) => Multiply both top and bottom with (a1-b2*i) to get rid of complex number in denominator. [(a1+b1*i) *(a2-b2*i)] / [(a2+b2*i)*(a2-b2*i)] = (a1*a2 + b1*b2) + (a1*b2 - a2*b1)*i  / √(a2^2 + b2^2)

Z = rCosθ + i* rSinθ (in polar form for any complex number). It's also written as Z = rCisθ or r∠θ

It's much easier to multiply and divide polar numbers by using polar form.

Z1*Z2 =  ( r1Cosθ1 + i* r1Sinθ1 ) * ( r2Cosθ2 + i* r2Sinθ2 ) = r1*r2 [ ( Cosθ1.Cosθ2 - Sinθ1.Sinθ2) + i (Sinθ1.Cosθ2 + Cosθ1.Sinθ2) = r1.r2 [ Cos(θ1+θ2) + i Sin(θ1+θ2) ] => In polar form, magnitude multiply while angles add.

Similarly Z1/Z2 = ( r1Cosθ1 + i* r1Sinθ1 ) / ( r2Cosθ2 + i* r2Sinθ2 ) => Multiply both top and bottom by ( r2Cosθ2 - i* r2Sinθ2 ) to get rid of complex number in denominator => r1/r2 [ ( Cosθ1 + i* Sinθ1 ) * ( Cosθ2 - i* Sinθ2 ) ] / ( 1 )] = r1/r2 [ ( Cosθ1.Cosθ2 + Sinθ1.Sinθ2) + i (Sinθ1.Cosθ2 - Cosθ1.Sinθ2) = r1/r2 [ Cos(θ1-θ2) + i Sin(θ1-θ2) ] => In polar form, magnitude divide while angles subtract.

We can extend above observation to n complex numbers, where angles add/sub and magituddes mult/div.

i.e Za1.Za2....Zan / (Zb1.Zb2... Zbn) = ra1.ra2...ran/(rb1.rb2...rbn) . Cis [  θa1+θa2...+θan - (θb1+θb2+...+θbn) ]

De Moivre's Theorem:

It's a special case of above, where power n of any complex number is Z^n = r^n Cis nθ

Link => https://en.wikipedia.org/wiki/De_Moivre%27s_formula

To find n-th roots of Z, we can use the above formula, where  Z^(1/n) = r^(1/n) Cis [ (θ+2πk)/n] where k is an integer from 0 to (n-1). As the fundamental theorem of Algebra says that there has to be n roots, we get n solutions. The term 2πk can be added to the angle of any complex number, but it remains the same number. But when we divide by n, then the angle 2π divided by n gives new angles within the 360 degrees yielding unique points.

Though De Moivre's theorem as written above is not true for non integer values of n, it was proved by Euler that it holds for even complex number n.

 If $z=\rho e^{i\theta }$ then $z^p={\rho }^p{e}^{i\theta p}$.where p is any real or imaginary number.

Proof is from Euler’s Theorem (which is very elegantly proved from Maclaurin Series), which says: $e^{i\theta }=\cos \theta +i\sin \theta$

 So, letting $r$be any real/complex number: $(\cos \theta +i\sin \theta {)}^r=(e^{i\theta }{)}^r=e^{i\left(r\theta \right)}=\cos \left(r\theta \right)+i\sin (r\mathrm{\theta )\; =>\; Hence\; this\; holds\; true\; for\; any\; complex\; or\; real\; value\; of\; exponent.\; The\; thing\; to\; realize\; is\; that\; this\; will\; give\; multiple\; solutions,\; while\; De\; Moivre\text{'}s\; Thm\; considers\; only\; 1\; solution\; when\; delaing\; with\; integer\; values\; of\; n.}$

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Published: Thursday, 06 July 2023 19:21

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States/Phases of matter:

There are three common states of matter: solid, liquid, and gas. (Plasma is another state of matter but is beyond this discussion.)

Libretexts has this excellent material on it: https://chem.libretexts.org/Courses/Bellarmine_University/BU%3A_Chem_104_(Christianson)/Phase_1%3A_The_Phases_of_Matter

A solid will maintain its shape (ice, wax, steel), while a liquid will flow and take the shape of its container (water, mercury). A gas or vapor will fill all available volume (steam, air, mercury). There is a 4th state which is plasma, also known as ionized gas state. It's a gas state where atoms lose their electrons in presence of very high electromagnetic field, which causes these electrons to start conducting electricity, which isn't typical of gases. In nature, lightning is the most common example of plasma. It's the most common state in the universe, since all stars are in plasma state. However, in real life we don't encounter this state, so we don't discuss the plasma state.

States of matter are generally based on the form of the substance at room temperature and pressure. Things like air are gaseous at room temperature while water is a liquid. When heated, water become gaseous, but is referred to as a vapor. At atmospheric temperature (25C) and pressure (1.0 atm), most of the elements are found in solid state. Mercury (Hg) and Bromine (Br) are the only 2 elements found in liquid state. All noble gases and H, N, O, F and Cl are the only ones found in gaseous state.

For Pressure details, see "Force, work, energy" section under Physics.

GASES:

Maxwell-Boltzmann distribution (MBD): Maxwell-Boltzmann (MB) distribution gives the distribution of speed of gas particles. It's derived from MB statistics (MBS) which s the more general case. MBD is a special case of MBS. MBD was found first by Maxwell in 1860, and then proved formally in 1870 by Boltzmann.

Link for MB statistics (not distribution) => https://en.wikipedia.org/wiki/Maxwell%E2%80%93Boltzmann_statistics

MBS describes the distribution of classical material particles over various energy states in thermal equilibrium. It states that # of particles, N_i with energy ε_i is:

where total # of particles N = Σ N_i and

Z = ∑ i g i e − ε i / k T ,

Basically what MBS is trying is getting the fraction of particles in a particular energy state (the numerator and denominator with T term in the exponents) and then multiplying by N to get total particles. Let's see if it makes sense. At 0K (absolute 0 temperature), the number of particles in any energy state is 0. This makes sense as at 0K, there are no energy states besides the ground energy state, and all particles are in that energy state. As Temp inc, the number of particles in higher energy states begin to inc.

MBD is a probability dist func of particles with a particular speed and applies to ideal gas. We can derive MBD from MBS by observing that  ε_i is the kinetic energy of that particle = 1/2*m*v^2, and then doing some manipulation.

Therefore, the Maxwell–Boltzmann speed distribution (i.e MBD) is :

f ( v ) = ( m 2 π k B T ) 3 / 2 4 π v 2 exp ⁡ ( − m v 2 2 k B T ) where m=mass of particle

The above eqn is of the form x*e^-x. It starts from 0 at x-=0 and goes to 0 at x=infinity. It's not a gaussian distribution as it's not symmetric. It's a chi distribution.

If we find RMS velocity of all particles, it's V = √ (v1^2 + v2^2 + ...). We use func above to find # of gas particles having speed b/w v and dv. We then integrate it as v^2*f(v)*dv, which leads to

V_rms  = √ (3*k_b *T)/m = √ (3*R *T)/M = Where M is molecular mass and R = k_b * Na (universal gas constant explained below)

V_rms of diatomic N₂ gas at 300K = sq rt[ 3*1.38*10^-23*300/(1.67*(10^-27)*28)] = 515m/s. Total KE of each particle = 1/2*m*(V_rms)^2. = 3/2*k_b *T. What this implies is that KE of any gas is the same for a given temp (doesn't depend on anything except Temp).

From above eqn, V_rms is higher for lighter gas, and lower for heavier gases. With pdf plotted on Y axis and rms velocity plotted on x axis, pdf curve's peak has to be at lower velocity for heavier gases. However total kinetic energy of any gas is dependent only on Temp. Since RMS velocity became lower for heavier gases, more particles will need to within the vicinity of the peak to keep energy same (i.e area under the curve can't change). This causes the curve to start shifting right for lighter gases (i.e curve starts flattening for lighter gases).

Total Kinetic energy of 1 mole of gas = 1/2* (3*k_b *T) * Na = 3/2*R*T. This is the most known eqn that KE of any gas at any Temp is 3/2*R*T (which is only dependent on Temperature and nothing else). This can also be seen from the observation that for a given thermodynamic system at temp T, the average thermal energy carried by each microscopic degree of freedom in the system is ⁠1/2⁠ kT (i.e., about 2.07×10⁻²¹ J, or 0.013 eV, at room temperature). Since particles in 3D have 3 degrees of freedom, each particle has 3/2 kT of energy. For 1 mole of particles, kinetic energy = 3/2kT*Na = 3/2RT = 3/2*8.3 J⋅K⁻¹⋅mol⁻¹ = 12.1 * 300K * 1mole =  3735 J = 3.7KJ.

Since KE in gas phase can only appear as multiple of 1/2RT, we should see that per mole per kelvin, we should see that it should be in integer multiple of 4.1J ⋅K⁻¹⋅mol⁻¹ .

NOTE: Each particle's KE is ~0.02eV, which is lot lower than the energy needed to excite electrons from one energy to another in any atom or molecule (which is in order of 1eV to 10eV). That's why heat can't excite electrons, but light waves can. However at very temp (> 10K temp in stars), thermal energy does cause electron excitation. 

Internal Energy (U) and degrees of freedom (DOF): Internal energy of a gas includes not only translational energy due to motion of particles, but also rotational energy (molecules spinning) and vibrational energy (molecules vibrating). All of these energy contribute to KE. On top of KE, we may also have PE resulting from proximity of particles. Combing both of these (KE+PE), we get Internal Energy (U) of the particle. What all motion (translational, rotational or vibrational) any molecule has is determined by it's degree of freedom.

Degree of freedom is the number of variables required to describe the motion of a particle completely. For any molecule with n atoms, each molecule has 3 degrees of freedom as the molecule is free to move in any of 3 dimensions. For n atoms in the molecule and assuming each atom within the molecule can also move independently in 3 dimensions, it will have a total of 3*n degrees of freedom. For ex, He gas will have 3 degrees of freedom, while N₂ gas will have 3*2=6 degrees of freedom.

However, all atoms within molecule are not free to move in any motion as these atoms are bonded together. The molecule as a whole will have 3 translational and 2 or 3 rotational motion (2 for linear molecule and 3 for non linear molecule). The remaining 3N-6 or 3N-5 will be vibrational motion. For non-linear molecules, all rotational motions can be described in terms of rotations around 3 axes, the rotational degree of freedom is 3 and the remaining 3N-6 degrees of freedom constitute vibrational motion. For a linear molecule however, rotation around its own axis is no rotation because it leave the molecule unchanged. So there are only 2 rotational degrees of freedom for any linear molecule leaving 3N-5 degrees of freedom for vibration. The sum total is always 3N (not sure why?). And to complicate matters, the degrees of freedom changes with Temp too !!

This link has few videos to explain => https://openbooks.library.umass.edu/toggerson-131/chapter/energy-associated-with-motion-at-the-molecular-scale-temperature-and-thermal-energy/

Each degree of freedom has 1/2RT of energy. For monoatomic gas such as Helium, atoms have translational energy only. So, total energy is 3/2RT. For diatomic gases such as Nitrogen gas (N₂) , on top of translational energy, we have rotational energy of RT due to 2 degrees of motion, while vibrational energy is almost zero. So, Total internal Energy U of N2 gas is 5/2RT. For other molecules, we have vibrational energy too, which may contribute to total energy. 

U in solids, liquids and gases: In gases, calculating total energy was easy as gas molecules only had KE (no PE as molecules don't interact with each other due to large distances), and hence their energies can be calculated in isolation. Any molecule whether in solid, liquid or gas form has KE in all forms, it's just that contribution from translation, rotational and vibrational motion changes, which changes total KE too. Gas form has the highest U and is mostly from KE due to translational energy (as particles are far apart, no interaction b/w particles, so potential energy negligible). U is lower in liquid form than gas as there's limited translational motion, but at same time PE is higher as particles are closer. Solids have the lowest U as particles are fixed in place, so only KE is from vibration of particles. However PE is highest as particles are closest together. At absolute 0K, KE of particles go down to 0, but still have some residual PE known as zero point energy.

PE change: PE at infinity is taken as 0. If same charge particles come closer together, then we have to apply energy to bring them closer, so their PE has to inc (since our work done gets stored as PE). If opposite charged particles come closer together, then they release energy as they come closer, so their PE has to dec (since we have to do -ve work which gets stored as PE). PE decreases as neutral particles come closer, as there are attractive forces b/w neutral particles (They behave as opposite charged particles or have attractive forces due to Van der waals (see below) in solid, liquid states). However, beyond a certain proximity, when particles get too close, then repulsive forces start dominating and PE starts to inc rapidly. We are talking about distances in solids/liquids where these attractive forces exist, so we can treat different particles as having opposite charges.

Latent heat during phase change: As a liquid changes to gas form, it takes in latent heat at it's boiling point, which doesn't inc the temperature. All of this latent heat goes into the liquid in increasing PE as particle distance increases. ,

Specific heat within a phase: Within a given state, specific heat is defined as heat needed to raise the temp by 1K for 1 mole or 1 g of the material. Specific heat is defined for constant pressure (C_p)  and constant volume (C_v) as it's different if pressure is held constant vs if volume is held constant. For solids and liquids, the 2 values are pretty close, but for gases they always differ by "R". C_p  > C_v as work is done to expand volume, on top of increasing the temp.

C_p - C_v = R = 8.3 J/mol-K

Proof: From ideal gas law (see below), PV=nRT. At constant Volume, all heat energy goes into raising the temp, while at constant pressure, heat energy goes into raising temperature as well as doing work to increase volume.

Work done by Pressure P in increasing volume by ΔV = P. ΔV (as force F is moving distance "d" resulting in work F.d. Using area, we get W = P.A.d = P. V

Energy supplied at constant pressure Q_p = Q_v + P. ΔV => Q_p = Q_v + nRΔT. => nC_pΔT = nC_v ΔT + nRΔT => C_p = C_v + R

Since internal energy (U) always goes in increments of 1/2RT, (i.e U = 1/2RT*degrees_of_freedom), this implies that specific heat of gases also goes in increments of 1/2R. From specific heat number @ constant volume, we can figure out the degrees of freedom in a given gas molecule (since at constant volume, all of the work done goes as internal energy and not into expansion work). 

U is the reason, why applying heat changes from solid to liquid to gas => The internal energy increases which changes the state (higher energy means more movement, so phase goes towards gaseous state). Specific heat of any material in the 3 states, gives us the amount of internal energy of any molecule in that particular state (assuming particle remained in that state all the way from 0K to the given temp).

Ex: Internal energy of Water calculated from specific heat: 

• Solid water (Ice) => specific heat = 36J/mole-K. So, at 0C (273K), Internal energy of ice = 36*273 = 9.8 KJ/mole.
• Latent heat from ice to water => 6KJ/mole. 100X more than specific heat of ice.
• liquid water (water) => specific heat = 75J/mole-K. So, at 27C (300K), Internal energy of water = 75*300 = 22.6 KJ/mole. 
• Latent heat from water to vapor => 44KJ/mole. 500X more than specific heat of water. The latent heat of vaporization for water changes with pressure. Different pressure cause water to boil at diff temp (see boiling pt section below). 
• gas water (vapor) => specific heat @ constant pressure (C_p) = 34 J/mole-K. So, at 100C (373K), Internal energy of vapor = 34*373 = 12.7 KJ/mole. specific heat @ constant volume (C_v) = 25J/mole-K. Water vapor has 7 degrees of freedom (3 each for translational, 3 for rotational and 0-1 for vibrational at lower temp).  U = 1/2RT*(degrees of freedom) = 4.1*7 = 25 J/mole-K which matches experimental value. The difference is R as expected.

FACT: water has the highest specific heat of 1 cal/g-K. This is how calorie was defined, where 1 calorie raises the temp of 1g of water by 1C. 1 calorie = 4.18J. 1 mole = 18g of water, so speciifc heat of water per gram = 75/18=4.2J/g-C, which matches with what it should be. Water's specific heat is the highest of most of the elements, and many compounds (metal's specific heat is in range of 20J/mole-K, which is 1/4th of water's. When we calc it in per gram, difference is even larger). Because of such large specific heat of water, it's able to take a lot of heat, w/o raising its' temperature by much, which is in turn able to cool earth. 

Ex: Internal energy of Noble gas Helium, which doesn't form any bonds with any other atoms.

• Solid He => specific heat = 2J/mole-K. He becomes solid only at very high pressure and Temp of 1K. So, at 1K, Internal energy of solid He = 1*2 = 2 J/mole.
• liquid He => specific heat = 20J/mole-K. He becomes liquid at 4K (at atm pressure). So, at 4K, Internal energy of liquid He = 20*4 = 80 J/mole.
• gas He => specific heat @ constant volume (C_v) = 12J/mole-K (or 12J/4g=3J/g-K). So, at 27C (300K), Internal energy of gas = 12*300 = 3.6 KJ/mole. Helium has 3 DOF. So U=1/2RT*3 =12.3J/mole-K.  specific heat @ constant pressure (C_p) = 20J/mole-K. 

As you can see, total U of any gas should be the sum of all energies that were put into it from solid state at 0K to it's gaseous state at some higher temp. When we used MBD to find the total energy of any gas, we assumed that the particle was always in gas form starting from 0K. Since that's not true, we have to add additional energy added during solid and liquid phases (incl latent heat) to bring it to gas phase. When we add all these energies, only then do we get the total energy in gas phase. 

Internal Energy vs atomic energy: This internal energy that we have been talking about is the thermal energy, This is energy for collection of atoms gaining energy due to heat. Other energy that we have within atom is in formation of atom itself => energy needed for formation of nucleus (nuclear energy) + energy needed for getting 2 electrons in it's orbit (Ionization energy or IE). These energies are orders of magnitude more than U. As an ex, IE for He is 7.6MJ (1000x more than U of He), while nuclear energy is 2.7BJ (1000x more than IE). These energies are also dependent on Temp? FIXME ?

• Ionization Energy (IE): Explained in Chemistry section on atoms.
  • Fermi Dirac Eqn: Probability that a energy state E is occupied by electrons is dependent on Temp and is given by FD distribution. So, higher Temp will get more electrons in higher energy states (i.e more easily removable from the atom)
• Nuclear energy (NE): Nuclear binding energy from proton and neutron combining to form a nucleus is very high. NE can be calculated by using Einstein's E=m*c^2 by observing the mass defect during the formation. For He nucleus (2 proton, 2 neutron), NE for 1 atom = 28 Mev. So, 1 mole has NE=2.7*10^9J. For more details, see in "nuclear chemistry" section.

Ideal gas law: This provides a relation between P, V, T of an ideal gas (ideal gas is one with no interaction among gas particles and assuming gas particles having zero size) with the number of gas molecules for an ideal gas. There were multiple laws discovered by various scientists, but they were all combined into one once "Ideal gas law" was found. Link: https://en.wikipedia.org/wiki/Ideal_gas_law

The law states that

P*V = n*R*T => where P,V,T are pressure (in pascal), Volume (in m^3), Temperature (in Kelvin) of gas and n=amount of gas particles (in moles), R= universal gas constant = (boltzmann constant) * (Avagadro constant) = 1.380×10⁻²³ J⋅K⁻¹ * 6.022×10²³ mol^-1 = 8.3 J⋅K⁻¹⋅mol⁻¹ . Here R represents energy in 1 mole of gas particles per unit temp.

The equation above is written in many different forms. The one above is the molar form. Under STP (T=273K, P=atmospheric pressure = 1 bar = 101kPascal), 1 mole of gas occupies V=nRT/P = 8.3*273/(101*10^3) m^3 = 0.0224 m^3 = 22.4 L (since 1L=1000 cm^3 = 1/1000 m^3). So, 1 mole of any ideal gas is always 22.4L in volume under STP. What is very surprising is that no matter what gas it is, it always occupies the same volume under same conditions of P, T. Whether the gas is water or oxygen or complex compound, they all occupy 22.4L for 1 mole of gas under STP. This doesn't seem intuitive, as one might expect gases to have different forces, size, etc and hence occupy diff volumes.

Derivation: Ideal gas law is easy to derive. See wiki link above. Simple proof:

Consider a container of Volume V with N gas particles moving with rms velocity v. Assuming random movement is equally likely in all 3 directions, x,y and z, 1/3 of the particles move in x direction (1/6 th move in +X, while 1/6 th move in -X dirn), strike against the container wall of Area S, and bounce back with reverse velocity v. Momentum change = 2mv for each particle, Force due to each particle=Momentum change/t = 2mv/t. For "q" particles hitting in time t, F=2mv/t*q. In time t, volume of particles striking the wall are = v*t*S. If there are N particles in Volume V, then number of particles in Volume v*t*S = v*t*S*N/V. But only 1/6th are striking. So, number of particles striking wall in time t = v*t*S*N/V *1/6. This is "q". So, P=F/S = 2mv/t*q*1/S = 2mv/(tS)*1/6*v*t*S*N/V=1/3m*v^2*N/V. So, PV=1/3Nmv^2

Using Maxwell-Boltzmann distribution above, ,we found rms velocity (v^2) which comes out to 3k*T/m. So, PV=1/3N*m*3kT/m=NkT => PV=NkT. This is the molecular formula of ideal gas law. If we take n moles of gas, then PV=n*N_A*kT => PV=nRT

Deviation from ideal gases: Real gases follow ideal gas laws only at low pressure and at high temperatures. This happens because at low pressure molecular volume of gases is still negligible compared to volume of container. Also, intermolecular forces are low amongst the particles. The same thing happens at high temp, where particles are able to break the intermolecular forces and don't interact with each other. Van der waals eqn which has experimentally determined constants a,b corrects the ideal gas law to incorporate it for real gases. Plots are shown in link above (section 1.7). When very high pressure or low temps are applied, gases change state to liquid, which is an extreme case, where intermolecular forces are very high, and KE is not enough to break them apart.

Dalton's Law of partial pressure: It states that total pressure of a mixture of ideal gases is equal to the sum of the partial pressures of the component gases. i.e if 2 gases are in their own containers with pressure P1 and P2, then, they are both put in the same size container, then total Pressure = P1+P2, as long as the gases don't react with each other. It seems counter intuitive, as we would expect presence of other gas to affect pressure of 1st gas compared to when it was by itself. Apparently that doesn't happen. Reason is that collisions are elastic, so presence of any amount of gas particles doesn't affect the energy of our gas particles (it's only dependent on Temperature), so energies of all gas particles can just be added up.

Phase changes: When a substance changes from one state of matter to another, it experiences a phase change. For instance, ice melting into water is a phase change from solid to liquid. Water changing into steam is a phase change from liquid to vapor. Energy in the form of heat is required to create a phase change. The elements in periodic table change state, as well as the compounds formed from them. Infact most matter made up of any number of compounds has to exist in all of the 3 states. It's just that at room temperature and pressure, it exists in one of the 3 states. But by changing temp and pressure, it can be converted to other states too.

Section 3 from above link (dedicated to phase changes only) => https://chem.libretexts.org/Courses/Bellarmine_University/BU%3A_Chem_104_(Christianson)/Phase_1%3A_The_Phases_of_Matter/3%3A_Phase_Changes

Intramolecular Bonds:

We saw intramolecular bonds under "atomic bonds" section. These are bonds within atoms of a molecule. Ionic, Covalent and metallic bonds fall in this category. 

Intermolecular Bonds:

Intermolecular forces (IMF) are forces that exist between molecules. They are much weaker than intramolecular bonds which are atomic bonds (explained under atomic bonds). IMF are important because they determine the physical properties of molecules like their boiling point, melting point, density, and enthalpies of fusion and vaporization. IMF are called weak bonds, while atomic bonds are called strong bonds. The force within a molecule to break the bonds is lot higher at 430 KJ/mole, while IMF is lot lower at 17KJ/mole (for conversion from gas to liquid) for HCl. So, these IMF are lot easier to break, and hence change phases. The phase in which a substance exists depends on the relative extents of its intermolecular forces (IMFs) and the kinetic energies (KE) of its molecules.

Intermolecular forces: https://chem.libretexts.org/Bookshelves/General_Chemistry/Map%3A_Chemistry_-_The_Central_Science_(Brown_et_al.)/11%3A_Liquids_and_Intermolecular_Forces/11.02%3A_Intermolecular_Forces

All of these attractive IMF are also called "Van der Waals" forces, after the name of the scientist who found these. There are 3 kinds of van der waals forces.

1. Dipole-Dipole attraction: Polar molecules such as HCl have +ve charge on one side and -ve charge on other side resulting in a dipole. An attractive force between HCl molecules results from the attraction between the positive end of one HCl molecule and the negative end of another. This attractive force is called a dipole-dipole attraction.
   
   1. Effect of dipole dipole attraction is apparent when we compare Fluroride gas with HCl. Both have same molecular mass, but polar nature of HCl causes the molecules to stick together, while F₂ molecules being non polar don't have this attractive force. As a result, HCl becomes gas at 188K,, compared to F₂ molecules which boil at much lower temp of 85K.
2.  London dispersion forces : These forces are present in all matter. London dispersion forces are weak attractions between molecules. They can occur between atoms or molecules of any kind, and they depend on temporary imbalances in electron distribution. This causes formation of temporary dipoles, resulting in electrostatic attraction b/w molecules. Being very weak, they become significant only when the molecules are very close. Larger and heavier atoms and molecules exhibit stronger dispersion forces than do smaller and lighter atoms and molecules. In a larger atom, the valence electrons are, on average, farther from the nuclei than in a smaller atom. Thus, they are less tightly held and can more easily form the temporary dipoles that produce the attraction. The shapes of molecules also affect the magnitudes of the dispersion forces between them. Compact shapes provide less surface area and hence less force, while elongated shapes provide a greater surface area available for contact between molecules, resulting in correspondingly stronger dispersion forces.
   1. Melting point (MP) and boiling point (BP) trends in periodic table:
      1. MP and BP have high correlation, they trend together, i.e elements with high MP also have high BP.
      2. As atoms become bigger in size (going down the periodic table), MP and BP inc (as dipoles are easier to form, resulting in higher van der waals forces). s1, s2, p1 and p2 columns don't show this trend (or show the trend in reverse)
      3. Carbon has highest MP at 3500C, and depends strongly on the form (diamond form has highest MP). Carbon also has high BP at 4800C, but Tungsten has the highest BP at 5500C.
      4. Helium has the lowest MP at 1K  and lowest BP at 4K (basically it exists in gas form only, it never exists in solid or liquid form except close to absolute 0)
3. Hydrogen Bond: This is a a bond that is usually b/w +ve charged H atom and another -ve charged atom.  This bond forms due to EN diff b/w the 2 diff atoms that form the molecule. Water is one of the ex of H bond. Even though they are called, they are not real bonds.
   1. In general, these are the requirements for a H bond:
      1. A polar covalent bond needs to exist b/w H atom and a highly EN atom as N(2p³), O(2p⁴), and F(2p⁵), These are the only 3 that have large EN difference to form a highly polar bond.
      2. There should be at least one active lone pair of electrons available in highly EN atom. Lone pairs at the 2-level have electrons contained in a relatively small volume of space, resulting in a high negative charge density.
   2. Once these 2 conditions are satisfied, a bond starts forming b/w lone pair of electrons in 1 molecule which is highly EN with highly positive hydrogen atoms of another molecule. This bond has 1/10 the strength of an avg covalent bond, so it is strong enough to change properties of that molecule.
   3. NH₃, H₂O and HF are 3 examples of compounds which form strong enough H bonds that their boiling points are lot higher, compared to what is expected in the absence of H bond.
      1. Water is the perfect ex of how H bond causes higher boiling pt. There are 2 pair of lone electrons available on Oxygen to form 2 H bonds with neighboring Hydrogen atoms of another molecule. Even though H2O has covalent bonds and hydrogen and oxygen atoms share their electrons, they end up developing a polarity due to higher electronegativity of oxygen compared to hydrogen. Oxygen ends up getting a slightly negative charge, while hydrogen a slightly positive charge. This allows different molecules of H2O to form a lattice structure with -ve polarity of oxygen of 1 molecule forming polar bonds with +ve polarity of hydrogen atoms of another molecule. These bonds are called Hydrogen bonds (and NOT polar bonds). When temperature are low, the vibrational energy of each molecule is very low, and not enough to break these hydrogen bonds between hydrogen and oxygen of different molecules. This makes it a solid, where the molecules can't slide past each other. But as temperature rises, the vibrational energy of each molecule increases, causing these hydrogen bonds to get weaker to a point where the molecules can slide past each other. This forms a liquid. If we keep on increasing the temperature, these hydrogen bonds break completely free, and different molecules become independent of each other. This forms the vapor or gas state. This video from Khan Academy explains it => https://www.khanacademy.org/science/chemistry/states-of-matter-and-intermolecular-forces/states-of-matter/v/states-of-matter

Liquid -> Gas (phase change): Just like we have distribution of gas particles with different speed/energy (as per Maxwell-Boltzmann distribution), the same law applies to liquid particles too. Some liquid particles will have higher kinetic energy than others. Ones which exceed the energy needed to break off the liquid surface (and near the surface traveling upward) will leave the liquid surface and become gas. This is called evaporation. If this gas is allowed to escape, then avg energy of liquid particles will drop (dropping the temp). This will force absorption of heat from surroundings, which will maintain the avg speed of molecules in liquid. Over time, all liquid will evaporate. However, if the liquid is kept in closed container, then evaporation will stop after some liquid has evaporated. This is due to the fact that vapors that form above the liquid start exerting a pressure, and will start getting recaptured at the surface (i.e condense into liquid state again). The 2 rates => rate of evaporation and rate of condensation balance each other out. The pressure of vapor at which this happens is called "vapor pressure (VP)" and is constant for a given liquid at a given temperature (irrespective of it's volume). Vapor pressure of water at 25C is 24mm Hg (3 kPa). See chart below.

The magnitude of the vapor pressure of a liquid depends mainly on two factors: the strength of the forces holding the molecules together and the temperature. Lower the intermolecular forces, easier for molecules to escape as vapor, and hence higher the vapor pressure. Also, as Temp inc, more molecules have higher energy and can get to vapor state, and hence higher vapor pressure. VP inc exponentially with temp.

Boiling point: Boiling pt is related to the vapor pressure. We saw above that Vapor pressure of water at 25C is 24mm Hg (3 kPa). At this temp, water converts to gas only at surface (not thru out the liquid). This process is called evaporation and heat needed for this is called "latent heat of evaporation". If we keep increasing temp to increase the vapor pressure, then at some temp, the vapor pressure (gotten from the P-T chart) will exceed the atmospheric pressure (Assuming container open to atmosphere). As soon as this happens, water will start forming bubbles even deep within the liquid, indicating the bonds have been broken inside the liquid too. These bubbles will come to surface and escape. Basically any heat given to the liquid to boil is the latent heat of vaporization, and is needed during any conversion from water to vapor (both during evaporation and boiling).

How does boiling differ from evaporation? 3 ways:

1. Evaporation happens only at the surface, while boiling happens thru out the liquid.
2. Evaporation happens at any temp, while boiling only at fixed temp.
3. During evaporation, we can keep providing more heat, which will raising the temp of liquid + vapor. However, during boiling, temp doesn't inc until all the liquid gets converted into vapor. Boiling can be thought of as vigorous evaporation.

Effect of pressure on latent heat of vaporization (L_v) : As pressure inc, boiling pt temp inc, which causes latent heat of vaporization to dec. This is due to the fact that molecules are already more energetic at higher temp, so less heat is needed to convert them to vapor. For water, L_v goes from 2.5KJ/g @0.01atm to 2.0KJ/g @10atm pressure.

Boiling occurs when ambient pressure (pressure of all gases including the vapor from the liquid evaporating/boiling) is less than vapor pressure. Why? Vapor pressure is the pressure at which everything is at equilibrium when enclosed in a container. The amount of liquid-> vapor conversion is same as amount of vapor->liquid. If placed in an open container, vapor escapes, so there's no extra pressure exerted by vapor (little extra pressure may be exerted by vapor right at the surface as not all vapors are swept away). So if we get air pressure to be less than this vapor pressure in an open container, then water molecules throughout can break the bonds and exit the liquid (since that vapor pressure was needed to maintain equilibrium b/w liquid and vapor). Water will continue to evaporate as it needs to maintain the vapor pressure. This process will continue indefinitely, resulting in creation of bubbles in the liquid, which are actually vapors at vapor pressure. These will keep escaping, until all the liquid has escaped as vapor. This vigorous process is called boiling and the temperature at which it happens is called "boiling point" at that air pressure. For our definition, we define "Boiling point" as temp at which vapor pressure equals atmospheric pressure. At 100C, vapor pressure of water is 1 atm, so that is our boiling pt for water (at atm pressure).

Let's consider 2 cases of closed container:

1. First we make vacuum in a closed container with some water in it, at room temp. Then water will immediately start boiling as there's 0 ambient pressure, and at 25C, VP=0.02 atm. Ambient pressure of 0 < VP of 0.02 atm, so boiling starts. As it boils, water will reduce in volume as it gets converted into vapor. This vapor will start exerting pressure, and water will keep boiling until the ambient pressure (pressure due to vapor only) in container becomes equal to the vapor pressure at that temp. Then water will stop boiling. At this point, water is at it's boiling pt of 25C at 0.02 atm pressure. If we again heat water, then water should ideally again start boiling as it's right at it's boiling temp. Or it may look like it's evaporating (hard to say). From this point on, as more water get converted into vapor, it causes more pressure. It rises to get to VR at the new temp and then stops rising. As heat keeps getting added, VR will keep rising and water will always remain at it's boiling pt. VR keeps rising exponentially in accordance with P-T chart. Boiling pt of water is also going up all this time. Once all the water has evaporated or boiled, only then will the gas law take effect and pressure will start rising linearly with temp (as P/T remains constant). 
2. Now we start with closed container that has air in at atm pressure, at room temp. VP is 0.02 atm, so some water will evaporate, but no boiling as ambient pressure >> VP. If we start heating water, then VP will start inc, as more water gets converted into vapor. However no boiling will ever take place as ambient pressure is always 1 atm greater than the VP (from P-T chart). So, in presence of any other gas in the container, water will never boil no matter what the temp is, as boiling pt will always be at a higher temp than the current temp.

In an open container, VP doesn't actually Another way to think about boiling is this => Let's say we removed the walls of the closed container (in which there was vacuum) and replaced it with air pressure of 24mm Hg. Then the situation is still the same as that of a closed container. Now, we slightly decrease the air pressure. Then vapor molecules will start moving from high pressure to low pressure, creating a pressure lower than vapor pressure near the water surface. This will cause more water to evaporate 

Vapor pressure Vs Temp graph shows exponential relation, and do show that as Temp inc, Vapor pressure inc. So, at lower atm pressure, vapor pressure will equal atm pressure at lower temp. So liquids boil at lower temp, when atm pressure is reduced.

Water Vapor Pressure chart => https://en.wikipedia.org/wiki/Vapour_pressure_of_water

Melting point: Just as boiling pt is defined, melting pt is defined as Temp where solid to liquid conversion happens.

Phase change graph: Pressure and Temp are the only 2 var affecting phases of any material. Graphs depicting these phases with P, T being on X,Y axis are called Phase change graphs.

Critical Temperature/Pressure: Once a gas is above its critical temperature and critical pressure (i.e critical point), it is impossible to get it to separate into a liquid layer below and a vapor layer above no matter how great a pressure is applied. Increasing the pressure only leads to the transition from gas to supercritical fluid.

Triple point: This is a particular temp and Pressure, where all 3 states exist simultaneously. For water, this point is at T=0.01C and P=0.006atm.

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Published: Tuesday, 27 June 2023 14:13

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Digestion and Respiration

Digestion and Respiration are the most activity our body in involved in to keep our body cells alive. We have a separate section about human cells.

Bunch of good videos on Youtube. These are ones I found helpful:

Respiration: Respiratory system consists of taking air from the nose all the way to the lungs an then back out from the nose.

• Introductory videos showing Respiratory system anatomy by Taim Talks Med 95 videos) => https://www.youtube.com/playlist?list=PLmzZnYRTmRK9W9q6j4YTeucpJ9dHsXlzV
• Good Anatomy video showing various cross section of nasal cavity by Dr Adel Bondok => https://www.youtube.com/watch?v=j4_lhFcemww
• Video showing 4 sinuses with real human skull and models (by Sam Webster) => https://www.youtube.com/watch?v=nLDEt0JzyyI

Digestion: Digestive system consists of taking food via the mouth, all the way to stomach and then excreted out.