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0:00
Every matter in this universe is just a combination of atoms. But atoms are not as simple as we think. They have their own rules, which are far beyond our understanding. However, since the early decades, we've been trying to understand what exactly an atom is. Bohr’s atomic model was a significant step in the understanding of atomic structure. But once we understood the concept of the wave nature of particles, the atomic model became more accurate and complex.
0:33
This trustworthy model is called a ‘Quantum model of the atom’. An atom is a combination of protons, neutrons, and electrons. Protons and neutrons are together in the center of the atom and are called the nucleus. The complex structure of an atom does not arise from the nucleus, but from the electrons. The tiny particles that orbit around the nucleus.
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Electrons orbit the nucleus in different orbits are shells. The shells are circular paths for electrons. These shells are fixed, meaning electrons can only exist within these shells. The first shell closest to the nucleus is denoted as n=1. The second shell, as n=2. The third shell n=3 and so on. The formula for calculating the maximum number of electrons a shell can hold is 2n squared,
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so the first shell can hold up to 2 electrons, the second shell up to 8, the third shell up to 18, and the fourth shell up to 32. This is known as bohr's model of electron configuration. These shells are located at specific distances from the nucleus. In a Hydrogen Atom, The first shell is located 0.53 Angstroms from the nucleus, and the second shell is located 2.12 Angstroms from the nucleus. These radii are fixed. The energy of an electron depends directly on its shell,
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E.g., in a hydrogen atom, an electron in the first shell has an energy of -13.6 eV, while an electron in the second shell has an energy of -3.4 eV, and So on. Bohr even found the velocity of the electron. from his equation, we can find that the electron in the 1st shell of a hydrogen atom, orbits the nucleus at a speed of 2.186 * 10^6 m/s. This means electrons orbit the nucleus at a fixed distance, fixed speed and fixed energy. Everything is quantized or has a fixed value.
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Even though it is a fine model for understanding the structure of an atom, it couldn't explain how atoms combine to form molecules through chemical bonds. Additionally, two important developments in our understanding of subatomic particles, raised many questions about Bohr’s model. So a more accurate model was required to solve the issue.
3:35
In 1924, French physicist Louis de Broglie proposed that every object has wave nature when in motion, a concept called the ‘Dual Behaviour of matter’. According to De Broglie, the formula for finding the wavelength of an object is Wavelength = h / m * v. E.g., imagine a car with a mass of a 1000 kgs, traveling at a speed of a 100 km/h. If the statement ‘Every object in motion exhibits wave nature’ is true, then the car should possess wave nature.
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By plugging these values into the formula, we get a wavelength of 2.385 * 10 ^ -38 m. This value is Ridiculously small, Trillions of times smaller than the size of a nucleus, and practically impossible to detect. So the wave nature of macroscopic objects is negligible. While the wave nature is negligible for everyday objects like a car, it becomes significant for tiny particles like an ‘Electron’. Consider the electron. It has a rest mass of only 9.1 * 10 ^ -31 kgs.
4:50
And Bohr’s Model gives the electron velocity in the first shell as 2.19 * 10 ^6 m/s. In this case, the wavelength of an electron is around 10 ^ -10 m, which is experimentally detectable, since the size of a hydrogen atom is approximately 10^ -10 m, and it has only one electron, that one electron can spread through the entire atom. It is truly amazing and confusing how a single electron can spread out through the entire atom. This is the wave nature of matter.
5:28
In particle form, We can pinpoint the location of an object, but when it's in wave form, how can we measure its location? The wave nature of particles is not only formulated through mathematical calculations, but also proven by experiments such as double-slit experiment, electron diffraction and the Davisson-Germer experiment. The double slit experiment is a fundamental demonstration in quantum mechanics, that reveals the wave particle duality of matter.
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The experiment involves an ‘electron gun’ that fires electrons, a ‘barrier’ with two narrow slits and a screen behind the barrier where the electrons land. Imagine a gun that fires balls instead of electrons. If we fire the balls toward the slits, some will bounce off the wall while others will pass through the slits. If we place a screen behind the slits, the balls that pass through will hit the screen.
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If we mark the locations where the balls landed, we will get a pattern of two strips aligned with the slits positions. But what happens when we fire electrons? If electrons were purely particles, we would expect a pattern similar to that of the balls. However, the results are unusual. When the wave passes through both slits, it's essentially splitting into two new waves. These two waves then interfere with each other. At some points, they cancel each other out.
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And at others, where peak meets peak, reinforcing each other. If we mark the locations on the screen, we observe a series of interference stripes at specific intervals. Similarly, when we shoot electrons through the slits, we see a wave like interference pattern on the screen. Surprisingly, when we place a detector near the electrons path, the wave nature collapses, and electrons exhibit particle behavior. This shows that electrons have both particle and wave nature.
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According to bore's model, an electron orbits in a precise circular path, similar to how planets orbit the Sun. However, the wave nature of electrons suggests that a single electron is spread out. While talking about wave nature and quantum things, it seems very complicated to learn, but if you're like me, you might find that some of these topics are not complex anymore. That's why I'm a user of Brilliant. This video sponsor.
7:54
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8:56
In 1927, Werner Heisenberg formulated a fundamental principle of subatomic particles, known as the ‘Uncertainty Principle’. He stated that it is impossible to measure both the position and momentum of a subatomic particle simultaneously. To understand why, let's first examine Bohr’s model of the hydrogen atom. In Bohr's model, the electron orbits the nucleus at the first energy level at a distance of 0.53 angstroms,
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and moves at a velocity of 2.19 * 10 ^ 6 m/s, with the mass of the electron being 9.1*10^-31 kgs, We can calculate its momentum. Momentum is mass * velocity. Thus, Bohr accurately calculated both the momentum and position of an electron, meaning both properties were known simultaneously. However, Heisenberg proposed a different perspective and questioned these precisely predicted values, and there is a reason for this. Consider the electron as a particle orbiting the nucleus.
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If we want to measure its position, a high energy photon must collide with the electron. A high energy photon has a shorter wavelength and greater energy, so when it collides with an electron, we can determine the electron’s position more precisely. This is because a shorter wavelength photon can resolve smaller details. When a photon collides with an electron, we can pinpoint the collision point, allowing us to know where the electron was during the interaction.
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This lets us determine the electrons position. However, the collision changes the electrons momentum. It alters the electrons velocity. So in this scenario, we can't accurately measure the electron’s momentum at the moment of the collision. This leads to uncertainty in the electrons momentum. Similarly, when a low energy photon collides with an electron, we can determine the electron’s momentum, but it becomes impossible to accurately determine its position.
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And also, high energy and low energy photons can't collide with a single electron at the same time. This means that at any given time, we can only measure either the electron’s position or its momentum, but not both. It's important to note that, this is not due to limitations in our measurement tools, but rather a fundamental property of nature. In nature, we are never able to precisely determine both properties of a subatomic particle simultaneously.
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This principle shows that subatomic particles have both particles and wave characteristics. This duality is why we can't determine both properties at the same time. Although we can't determine their precise values, we can make approximate predictions. These approximations are expressed as probabilities. ., Precise statements about the position and momentum of electrons must be replaced by statements of probability. This probability framework leads to the ‘Quantum model of atoms’.
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To support and better understand the wave nature of matter, a more advanced mathematical framework was needed. In 1926, Erwin Schrödinger provided a mathematical formulation describing how electrons behave in a quantum system. In this case, our quantum system is an atom. Now let's revisit the wave form of the electron in the hydrogen atom. Here, the electron exists as a wave form all around the nucleus, forming what is known as an ‘Electron cloud’.
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It's important to note that these electron clouds are not like classical gas clouds. They are representations of the wave function. Within this electron cloud, the electron can theoretically be detected anywhere within its wave function. However, the electron is most likely to be found in specific regions. E.g., Imagine a ‘male dog’ roaming around your city all day. After observing him for a month, you notice that while he explores various parts of the city,
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he spends most of his time in a particular location, where a ‘female dog’ is located. So it's clear that the dog can be found anywhere in the city, but the highest probability is near the ‘female dog’. If you want to find the male dog, you don't need to search the entire city. Just stay near the female dog, and you'll have the best chance of finding him. Similarly, in the case of an atom, the electron can theoretically be found anywhere around the nucleus.
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However, most of the time, it'll be found in certain regions. These likely regions are described by the Schrödinger Equation. The Schrödinger Equation decodes the probability of where an electron is most likely to be found. It describes the probability density of an electron within an atom. Simply put, it shows where the electron density is higher. That is where the electron is more likely to be located. To understand this, consider a one dimensional standing wave.
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Imagine one edge of the wave starts at the nucleus and extends to infinity. At the nucleus, the probability of finding an electron is zero, because the wave’s Amplitude is zero there, the string doesn't move at the nucleus. As the distance from the nucleus increases, the wave’s Amplitude increases, and so does the electron density. Thus the probability of finding the electron is highest, where the wave has greater amplitude.
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At the peak of the wave, the probability of finding the electron is at its highest. As we move toward infinity, the probability of finding the electron gradually decreases, and at infinity it becomes zero. If we apply this standing wave concept and the Schrödinger equation to the Hydrogen atom, we find that 90% of the time the electron is likely to be detected near Bohr’s atomic radius. Unlike bore's model, where the electron follows a fixed orbit of 0.53 Angstroms,
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In the quantum model, the electrons position is spread out around the radius. The peak probability of finding the electron remains at the same distance as Bohr’s radius - 0.53 Angstroms, But there is a 10% chance that the electron could be found in regions closer to the nucleus or even further out. So overall, the electron density gives the atom a cloud like appearance. This diagram represents where the probability of finding the electron is highest.
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From this, we see that the probability density is greatest near Bohr’s radius. But this is a quantum model, and it doesn't just tell us only where an electron can be found, But it gives a proper description of the electrons behaviour. When we solve the Schrödinger equation for an atom, a new set of values emerges to explain electron behaviour. These values are known as quantum numbers. Quantum numbers provide critical information about an electron in an atom,
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such as its Size, Position, Shape, Energy, Orientation, and Spin.
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A shell is the home for electrons, and the ‘Shell number’ is like the ‘House number’. It represents where electrons can be found, and the size of the electron shell, denoted by the number ‘n’, called the ‘Principal Quantum Number’. This number is similar to Bohr’s model of orbits, but instead of fixed orbits, it determines the size of the electron cloud. The first shell is n=1, the second shell is n=2. The third is n=3, and so on. These ‘n’ values are always positive integers 1,2,3,4 etc, etc,.
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As the shell number increases, the size of the electron cloud increases, making the electron cloud larger. In other words, the highest probability of finding an electron is farther from the nucleus. Now, let's return to the Bohr’s model of electron arrangements In an atom. The number of electrons that can occupy each shell is given by the formula 2n squared. This means the first shell can hold up to 2 electrons.
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The second shell up to 8 electrons, the third shell 18 electrons, and the fourth 32 electrons and so on. The number of electrons increases as the shell number increases, Because increasing shell number means increasing in size, it can hold more electrons. This electron arrangement appears simple, but it is more complex than it seems. While it's true that the number of electrons held in each shell is exactly equal to Bohr’s model,
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the quantum model introduces an additional subdivision called ‘Sub shells’. Each shell is divided into sub shells, denoted by the ‘Azimuthal Quantum number’, or Orbital Quantum number ‘l’ If we consider shells as houses, sub shells are the floors. There are four major sub shells: s,p,d and f the first shell has only one sub shell ‘s’, the second shell has two sub shells: ‘s’ & ‘p’ the third shell has three: ‘s’,’p’ and ‘d’. The fourth shell has four: ‘s’,’p’,’d’ and ‘f’.
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The Principal quantum number of shells starts at 1 and continues with positive(+) integers, but the Azimuthal Quantum number for each shell takes values from 0 to (n-1). To make this easier to understand consider the house analogy again. The house number is the shell number. And each house has floors. House No:1 (n=1) is very small, so it has only one floor, which is a ‘s’ sub shell. So for the first shell, ‘l’ value is zero. House No:2 (n=2) is bigger than House No:1.
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So it has two floors, ground floor, which is sub shell ‘s’ and first floor, which is sub shell ‘p’. Quantum number of ‘s’ sub shell is 0, because it's a ground floor. And quantum number of ‘p’ sub shell is 1, because its first floor. Therefore, for the first shell, the ‘l’ value is 0. And for the second shell ‘l’ values are 0 & 1. House No:3 (n=3), is bigger than houses 1 and 2. So it has three floors : Ground floor, First floor and Second floor, corresponding to the ‘s’,’p’ and ‘d’ sub shells.
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So the third shell’s ‘l’ values are 0,1 and 2. As we already know, a floor is not the final part of a house. Similarly, in the Quantum model, a sub shell is not the final part of the atom.
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Floors can be divided into ‘Rooms’. Similarly, sub shells are divided into ‘Orbitals’. Orbitals are defined by the Magnetic quantum number ‘ml’. Before discussing the Magnetic quantum number, let's understand orbitals more clearly. Orbitals are rooms for electrons. This is the location where electrons are most likely to be found. The number of orbitals in a sub shell is given by the formula : 2l+1, where ‘l’ is the Azimuthal quantum number or the floor number.
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So the ‘s’ sub shell has one orbital, the ‘p’ sub shell has three orbitals. The ‘d’ sub shell has five orbitals, and the ‘f’ sub shell has seven orbitals. If we look at the house analogy again, we notice that increasing the floor number increases the number of rooms. Each orbital can hold two electrons. Now let's compare this with Bohr’s model of electronic configuration. According to Bohr, the first orbit can hold up to 2 electrons.
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In the quantum model, the first shell has only the ‘s’ sub shell, which has one orbital so it can hold 2 electrons. The second orbit can hold 8 electrons. In the quantum model, the second shell has two sub shells: s & p, the ‘s’ sub shell has one orbital, and the ‘p’ sub shell has three orbitals. Since each orbital can hold two electrons, the total number of electrons in the second shell is 8.
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The third shell has s,p&d sub shells. The s sub shell has one orbital, p has three and d has five orbitals giving a total of 9 orbitals and 18 electrons. From this we can see, that the number of electrons in each shell matches Bohr’s model. In Bohr’s model, electrons are fixed in orbits, but in the quantum model, electrons are described by a wave function, and we determine their location based on probability. So we need to understand in which region each electron is most likely to be found.
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In a hydrogen atom, there is only one electron, so it is easier to determine the probability of finding it, Since there is only one electron, the electron cloud surrounds the nucleus in all directions. To understand the position of electrons in higher level shells, we need to visualize the orbitals.
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The magnetic quantum number describes the orientation of orbitals within a sub shell, and is denoted by the symbol ‘ml’. The magnetic quantum number can take integral values from -l to +l, including Zero. For the s sub shell, the Azimuthal quantum number is zero. So the magnetic quantum number also. This value indicates a single orbital orientation, meaning there is only one possible orientation for the orbital which is spherical in shape and centered around the nucleus.
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The probability distribution of finding an electron is spherical in three dimensional space. The p sub shell has three orbitals, and its orbital quantum number is one. So the magnetic quantum numbers for the p sub shell are -1,0,+1. These numbers represent the positions and orientation of the orbitals, E.g., When we determine the position of something, we use an X-Y Axis Two dimensional diagram,
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which starts from zero and extends with negative numbers to the left of the center; and positive numbers to the right. These Magnetic quantum numbers function similarly by indicating the orientation of the orbitals in 3D space. The s sub shell orbital has a Spherical shape. But the p sub shell orbitals have a Dumbbell shape. The first magnetic quantum number of the p sub shell -1 represents the orientation along the axis of one of the three orbitals.
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We can consider the First orbital to be aligned along the Px axis in 3D space, meaning it extends along the X-axis with two ‘lobes’ on either side of the nucleus, forming a ‘Dumbbell’ shape. As the p sub shell has three orbitals, the other two are aligned along the Py and Pz axis. The Second orbital with a magnetic quantum number of +1 exists along the Y-axis, and the Third orbital is aligned with the Z-axis in 3D space.
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This dumbbell shaped region is where the electron is most likely to be found, but why is it ‘dumbbell shaped?’ This can be explained by one dimensional standing waves analogy. Electrons in ‘p’ sub shells have more energy than those in ‘s’ sub shells, which means they have a higher frequency. The nucleus is located at the center of the wave. While the wave oscillates, the part at the nucleus remains stationary.
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We already know that the non-oscillating part of a wave has a zero probability of containing an electron. On both sides of the nucleus, there is a probability of finding an electron. Therefore, the probability density is zero at the nucleus and higher on either side, creating the dumbbell shape. The location with zero probability of finding an electron is called a ‘Node’.
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The region with the highest probability is called a ‘Lobe’. In the case of the ‘p’ orbital, 2 lobes together form the dumbbell shape. Since each orbital can hold two electrons, this does not mean that each electron resides in a different lobe. Since two electrons in the same orbital share the same way function, they can exist in either lobe at the same time. The standing wave analogy is meant only for easy understanding. The node isn't simply based on frequency.
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In reality, it depends on the Azimuthal or angular quantum number. For the p orbital, the angular quantum number ‘l’ is one, so the number of angular nodes is one. From that we can see the dumbbell shape consists of two regions, lobes and nodes. Lobes are the areas with high probability of finding an electron, And the node is where the probability is zero. For a ’p’ orbital the nodal plane passes through the nucleus, dividing a single orbital into two lobes.
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The overall ‘p’ sub shell with its three orbitals looks like this. The ‘s’ Orbital has zero angular nodes, So the probability of finding an electron at the nucleus is not zero. The ‘d’ sub shell has five orbitals, and its orbital quantum number is 2, So its magnetic quantum numbers are: -2,-1,0,+1,+2 Because the orbital quantum number is 2, each ‘d’ orbital will have Two angular nodes and generally Four lobes.
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The five d orbitals and their orientations are: dxy : lies in the XY plane, with lobes between the X and Y axis. dxz : lies in the XZ plane with lobes between the X and Z axis. dyz : lies in the YZ plane with lobes between the Y and Z axis. dₓ²₋ᵧ² : lies in the XY plane with lobes along the X and Y axis. d𝓏² : oriented along the Z axis with a ring in the XY plane and lobes above and below. Higher sub shells have even more complex orbital shapes.
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Now take a carbon atom as an example. It has six electrons. The first two electrons fill the first shell, which has only one sub shell, which is ‘s’. The s sub shell has only one spherical orbital. So the first two electrons filled a 1s orbital, then we have four electrons remaining. Once the first shell is full, the electrons begin to fill higher energy levels. Starting with the Second shell, as we know, the second shell has two sub shells : s and p.
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the ‘s’ sub shell has one orbital and the ‘p’ sub shell has three orbitals. Two of the remaining four electrons fill the 2s orbital then we have the remaining two electrons. Once the 2s orbital is filled, the remaining electrons begin to fill the 2p orbitals. The P sub shell has three orbitals : 2px, 2py and 2pz, which can collectively hold six electrons However their remaining two electrons do not both occupy a single orbital. E.g. filling only 2px and leaving 2py and 2pz orbitals empty.
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Instead the two electrons occupy two Separate orbitals with one electron in each. But why don't the remaining two electrons pair up in a single orbital. To understand this need to look at one of the additional properties of the electrons.
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In 1925, Wolfgang Pauli proposed the ‘Pauli Exclusion Principle’, which states that no two electrons in an atom can have the same quantum state or quantum numbers. Eg., Consider the 2px orbital : the first Quantum number is the Principal Quantum Number (n=2). The second is the Orbital Quantum Number (l=1). And the third is the Magnetic Quantum Number (ml=-1). So the 2px orbital has three specific quantum numbers.
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We know that each orbital can hold two electrons, it means two electrons in a 2px orbital have the same quantum numbers. But since the Pauli Exclusion Principle says no two electrons can share the same quantum numbers then how can two electrons in a 2px orbital have the same set of quantum numbers? This is where the fourth quantum number comes into play The Spin Quantum Number denoted by (ms).
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In any orbital 2 electrons must have opposite spins, that means the two electrons in the same orbital will differ only in their spin. One electron has a spin quantum number of +1/2 and the other has -1/2. Since each orbital can hold only two electrons, these are the only two possible spin states. The property of a spin is a fundamental aspect of quantum particles. Now let's return to the electron configuration of a Carbon atom. The 1s orbital is filled first, followed by the 2s orbital.
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Two electrons remain, which will occupy one of the 2p orbitals : 2px, 2py or 2pz. However they do not pair up in a Single orbital initially. According to Hund's rule each of the three p orbitals gets one electron before any orbital gets a second. In the second shell, after the 2s orbital is filled, each of the 2p orbitals receives one electron before pairing begins. Electrons and singly occupied orbitals always have the same spin, Either spin up or spin down.
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This arrangement reduces electron - electron repulsion and makes the atom more stable. In an oxygen atom, which has eight electrons, four electrons are already filled in the 1s and 2s orbitals. Off the remaining four - three go into the individual 2p orbitals: 2px, 2py & 2pz. The fourth then pairs with one of the already occupied orbitals. Because all three 2p orbitals have the same energy, the final electron can pair with any of them. The paired electrons always have opposite spins.
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If we look into some elements from Lithium to Neon, we can clearly see how electrons fill orbitals according to energy levels and spin. From this, we can observe that electrons start filling orbitals in order of increasing energy. A higher shell corresponds to higher energy. The third shell has more energy than the second. However, sub shells within these shells complicate the arrangement. Take potassium as an example. It has 19 electrons. The first shell holds two electrons in the 1s orbital.
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The second shell holds eight electrons, two in the 2s orbital and six in the 2p orbitals. This leaves 9 electrons. The third shell can hold up to 18 electrons, two in the 3s orbital and six in the 3p orbitals. Now, one electron remains, and we might expect it would fill the 3d orbital, but the actual electron configuration of Potassium is not like this. Why? To understand this, we must look at model lungs rule.
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It states that electrons fill orbitals in order of increasing n+l values, where ‘n’ is the Principal Quantum Number, and ‘l’ is the Azimuthal Quantum Number. The ‘l’ values are as follows, 0 for s, 1 for p, and 2 for d. So for the 1s orbital, n=1, and l=0, giving an n+l value of one. For 2s, the value is 2. For 2p, n=2, l=1. So the value is 3. Similarly for 3s 3p and 3d : the n l values are 3,4 and 5 respectively. Now calculate the n+l value for the 4s orbital. n=4 and l=0, giving a value of 4.
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Compare this with the 3d orbital, which has an n+l value of 5. According to Madelung's rule, orbitals with lower n+l values fill first. So the 4s orbital is filled before the 3d orbital. This means electrons start filling the fourth shell even before the third shell is completely occupied. Therefore, the electron configuration of a Potassium atom is 1s 2s 2p 3s 3p 4s. The one remaining electron after the 3p orbital goals into the 4s orbital rather than the 3d,
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This electron configuration plays a major role in chemical bonding and the periodic table is organized based on electronic configurations. The quantum model of the atom explains chemical bonding between atoms provides a clearer understanding of subatomic particles behavior in quantum systems, and much more. However, our understanding of quantum systems remains incomplete due to their complexity and the probabilistic nature of quantum mechanics.
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We face challenges such as accurately predicting multiparticle interactions, understanding quantum entanglement, and bridging the gap between quantum and classical worlds. This means that the quantum model of the atom may NOT be the FINAL ONE.
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