Wave Nature of Matter Light has wave-like and particle-like properties Can matter have wave and particle properties? de Broglie s hypothesis: matter has wave-like properties in addition to the expected particle-like like properties Confirmed by electron diffraction experiments de Broglie proposed that electrons moving around the nucleus have wave-like properties Wave Nature of Matter Since electrons have wave-like properties, each electron has an associated wavelength λ = h mv where λ = wavelength h = Planck s constant m = mass v = velocity (mv = momentum) What is the wavelength associated with an electron of mass m = 9.109 x 10-28 g that travels at 40. 0 % of the speed of light? Wave Nature of Matter If all matter has wave-like properties (and an associated wavelength), why don t we notice it? If you run 15 km/hr, what s your wavelength? The wavelength is inversely proportional to mass. The wavelength of everyday objects is extremely small because its mass is large. 1
Wave Nature of Matter h= 6.63 10-34 J = 6.63 10-34 kg.m 2 s -1 Find out de Broglie wavelength (in m) of an electron of m=9 10-28 g; v= 1 ms -1 Α. λ = 7 x 10-4 B. λ = 7 x 10 14 C. λ = 7 x 10-29 D. λ = 7 x 10 5 Find out de Broglie wavelength (in m) of an electron of m=9 10-28 g; v= 5.9 10 6 ms -1 Α. λ = 1 x 10-4 B. λ = 1 x 10 14 C. λ = 1 x 10 10 D. λ = 1 x 10-10 Wave Nature of Matter h= 6.63 10-34 J = 6.63 10-34 kg.m 2 s -1 Find out de Broglie wavelength (in m) of a baseball of m= 142 g; v= 25.0 ms -1 Α. λ = 2 x 10-4 B. λ = 2 x 10 14 C. λ = 2 x 10-34 D. λ = 2 x 10-59 Find out de Broglie wavelength (in m) of the Earth of m = 6 10 27 g; v = 3.0 10 4 ms -1 Α. λ = 4 x 10-4 B. λ = 4 x 10 14 C. λ = 4 x 10 10 D. λ = 4 x 10-63 The Quantum Mechanical Model Waves don t have a discrete position! Spread out through space Cannot pinpoint one specific location Since electrons have wave-like properties, we cannot know their exact position, and velocity at any given time. Bohr s model ignores the wave properties of electrons 2
The Quantum Mechanical Model Macroscopic world: position and velocity (momentum) of a particle can be determined to infinite precision. Quantum Q mechanical world : an uncertainty associated with each measurement. Bohr s Model Bohr s model suggests that the electron has the lowest energy at Α. Furthest point from the nucleus B. Can not predict C. Close to the nucleus D. To be computed by the formula E=hν Bohr s model suggests emission of light occurs because Α. An electron jumps to a higher energy level B. An electron jumps to a lower energy level C. An electron jumps from the ground state D. Electrons collide with each other Bohr s Model An incorrect statement about n in Bohr s model is Α. It is called principle quantum number B. It does not relate to the energy of an electron C. Energy increases as n increases D. Stability increases as n decreases Bohr s model suggests absorption of light occurs because Α. An electron jumps to a higher energy level B. An electron jumps to a lower energy level C. An electron jumps from the ground state D. Electrons collide with each other 3
The Quantum Mechanical Model Heisenberg Uncertainty Principle: The exact position (location) and exact momentum (mass velocity) in space of an object cannot be known simultaneously. If you know the momentum of an electron, you can t know its exact location. Electrons don t move in well-defined circular orbits around the nucleus. In 1926 Schrödinger developed an equation that incorporates both the particle-like and wave-like behavior of electrons. Heisenberg s Uncertainty Principle Uncertainty in position Δx mδu h 4π Uncertainty in momentum Heisenberg s Uncertainty Principle Δx mδu h 4π An electron moving near an atomic nucleus has a speed 6 10 6 ms -1 ± 1%. What is the uncertainty in position ( Δx)? 4
The Quantum Mechanical Model Schrödinger wave equation describes the total energy of an electron in an atom based on its location and the electrostatic attraction/repulsion Solving the Schrödinger wave equation leads to a series of mathematical functions called wave functions (ψ) Although ψ does not have any yphysical meaning, the square of the wave function (ψ 2 ) describes the probability of finding the electron at a given location These wave functions are called orbitals and have a characteristic energy and shape. The Bohr model used a single quantum number (n) to describe an orbit, the Schrödinger model uses three quantum numbers: n, l, and m l to describe an orbital. An orbital: Orbitals and Quantum Numbers describes a specific distribution of electron density in space has a characteristic energy has a characteristic shape is described by three quantum numbers: n, l, m l can hold a maximum of 2 electrons A fourth quantum number (m s ) is needed to describe each electron in an orbital Orbitals and Quantum Numbers Principal quantum number (n): integral values n = 1, 2, 3, 4, describes the energy of the electron gy as n increases, the energy of the electron increases as n increases, the average distance from the nucleus increases as n increases, the electron is more loosely bound to the nucleus 5
Orbitals and Quantum Numbers Angular momentum quantum number (l): also known as the Azimuthal quantum number integral values l = 0, 1, 2, 3,.,(n-1) Example: If n = 3, then l = 0, 1, or 2. If n = 4, then l = 0, 1, 2, or 3. defines the shape of the orbital Orbitals and Quantum Numbers Angular momentum quantum number (l) The value for l from a particular orbital is usually designated by the letters s, p, d, f, and g: Value of l 0 1 2 3 4 Letter used s p d f g An orbital with quantum numbers of n = 3 and l = 2 would be a 3d orbital An orbital with quantum numbers of n = 4 and l = 1 would be a 4p orbital Orbitals and Quantum Numbers Magnetic quantum number (m l ): describes the orientation in space of the orbital integral values between l and -l If l = 1, then m l = 1, 0, -1 If l = 2, then m l = 2, 1, 0, -1,-2 6
Orbitals and Quantum Numbers When n = 3, the values of l can be Α. +3, +2, +1, 0 B. +3, +2, +1, 0, -1, -2, -3 C. +2, +1, 0 D. +2, +1, 0, -1, -2 When n = 4, the values of l can be Α. +3, +2, +1, 0 B. +3, +2, +1, 0, -1, -2, -3 C. +2, +1, 0 D. +2, +1, 0, -1, -2 Orbitals and Quantum Numbers When l = 2, the values of m l Α. +3, +2, +1, 0 B. +3, +2, +1, 0, -1, -2, -3 C. +2, +1, 0 D. +2, +1, 0, -1, -2 can be When l = 3, the values of m l can be Α. +3, +2, +1, 0 B. +3, +2, +1, 0, -1, -2, -3 C. +2, +1, 0 D. +2, +1, 0, -1, -2 Orbitals and Quantum Numbers What type of orbital is designated by n = 3, l = 2? Α. 3s B. 2p C. 3d D. 4s What type of orbital is designated by n = 4, l = 0? Α. 3s B. 2p C. 3d D. 4s 7
Orbitals and Quantum Numbers What type of orbital is designated by n = 2, l = 1? Α. 3s B. 2p C. 3d D. 4s What type of orbital is designated by n = 3, l = 0? Α. 3s B. 2p C. 3d D. 4s Summary Quantum Values Property Description number n 1,2,3. size shell l 0... n-1 shape subshell m l -l..0..+l orientation orbital The number of subshells in a shell = n The number of orbitals in a subshell = 2l+1 The number of orbitals in a shell = n 2 The shapes of Orbitals s-orbital- what s value of l? Spherical in shape the size of the s-orbital increases with increasing n 1s 2s 3s 8
Orbital Shapes Any nodal planes? Nodal planes are defined as those planar areas where the electron density is low Orbital Shapes p-orbital- what s value of l? What are the values of m l? Three p orbitals dumbbell shaped same size and energy within same shell (same n value) different spatial orientation m l = -1 m l = 0 m l = 1 Orbital Shapes Where are the nodal planes? 9
Orbital Shapes d-orbitals-what s value of l? What are the values of m l? five d orbitals are present in each shell where n > 3 same energy within same shell different shapes different orientation in space Where are the nodal planes? Orbital Shapes Third Shell Orbitals E n e r g y Shell n = 3 3p 3d 3s 10
What are the quantum numbers associated with the following subshells: Subshells n values l values m l values 1s 2p 3p 3d 5f Summary Compare and contrast the Bohr and quantum mechanical models Summary An electron has a 100% probability of being somewhere ORBITAL: The region in space where an electron is likely to be found The usual pictures of orbitals show the region where the electron will be found 90% of the time http://www.falstad.com/qmatom/ 11
Summary nodes: regions in space where the electron can t be found radial probabilities: the probability of finding an electron a certain distance from the nucleus Different orbitals have different shapes and sizes different energies 12