PHYS 213 Midterm Exam HKN Review Session Steven Kolaczkowski Alex Littlefield

First Law of Life Check your units

First Law of Thermodynamics Check Your Units d𝑈=𝑑𝑄+𝑑 𝑊 𝑜𝑛 =dQ−P𝑑𝑉 If volume is constant: d𝑈=𝑑𝑄

Entropy, Temperature, and the Second Law of Thermodynamics CHECK YOUR UNITS 𝑆≡ 𝑘 𝐵 ln⁡(Ω) where Ω is the number of microstates Ω 𝑡𝑜𝑡𝑎𝑙 = Ω 1 ∗ Ω 2 ∴ 𝑆 𝑡𝑜𝑡𝑎𝑙 = 𝑆 1 + 𝑆 2 𝑑𝑆= 1 𝑇 𝑑𝑈+ 𝑃 𝑇 𝑑𝑉− 𝜇 𝑇 𝑑𝑁 For an Isolated system Δ𝑆≥0 or Δ 𝑆 𝑈𝑛𝑖𝑣𝑒𝑟𝑠𝑒 ≥0 If Δ𝑆=0 the process is reversible Temperature: 1 𝑇 ≡ 𝑑𝑆 𝑑𝑈 𝑉,𝑁

Equilibrium and Counting Microstates We will observe the macrostate with the most microstates (maximum entropy) A microstate is an observable state Distinguishable: Ω= 𝑀 𝑁 Indistinguishable: Ω= 𝑀 𝑁 𝑁! 𝑆 𝑡𝑜𝑡𝑎𝑙 is maximized when T is constant across system (thermal equilibrium)

Heat Capacity and Latent Heat Heat capacity: the amount of energy required to raise the temperature of a material by 1K 𝐶 𝑉 = 𝑑𝑄 𝑑𝑇 = 𝐽 𝐾 𝑐 𝑣 𝑚𝑜𝑙 = 1 𝑛 d𝑄 d𝑇 = 𝐽 𝑚𝑜𝑙∙𝐾 𝑐 𝑣 𝑚𝑎𝑠𝑠 = 1 𝑚 𝑑𝑄 𝑑𝑇 = 𝐽 𝑘𝑔∙𝐾 𝐶 𝑣,𝑡𝑜𝑡 = 𝐶 𝑣,1 + 𝐶 𝑣,2 𝑑𝑄= 𝐶 𝑣 ∙d𝑇=𝑚 𝑐 𝑣 𝑚𝑎𝑠𝑠 ∙𝑑𝑇 = 𝐽 Latent Heat: The amount of energy needed to transition between states of matter T does not change during a phase change! 𝐿 𝑓 = Δ 𝑄 𝑠𝑜𝑙𝑖𝑑→𝑙𝑖𝑞𝑢𝑖𝑑 𝑚 𝑠𝑜𝑙𝑖𝑑 = 𝐽 𝑘𝑔 𝐿 𝑣 = Δ 𝑄 𝑙𝑖𝑞𝑢𝑖𝑑→𝑔𝑎𝑠 𝑚 𝑙𝑖𝑞𝑢𝑖𝑑 = 𝐽 𝑘𝑔

Ideal Gases and Equipartition CHECK YOUR UNITS Equipartition: The idea that thermal energy on average is distributed equally among all quadratic modes of motion <𝐾>= 𝑘 𝐵 𝑇 2 for each quadratic degree of freedom Monoatomic Gases: 3 Diatomic Gases: 5 Solids: 6 Mass doesn’t matter for calculating energy, but it does for finding average velocity Ideal Gas Law: Assuming that particles will not collide or interact with each other, only with the walls of the container, and are volumeless Works best with low densities and high temperatures (a few, fast moving particles) 𝑃𝑉= 2 3 𝑁<𝐾 > 𝑡𝑟𝑎𝑛𝑠 =𝑁𝑘𝑇=𝑛𝑅𝑇

Solids, More Heat Capacities, and Internal Energy CHECK YOUR UNITS 𝑈=𝑁<K> Ideal Solids 𝑈 𝑡𝑜𝑡𝑎𝑙 =𝑁 < 1 2 𝑚 𝑣 2 > 𝑡𝑟𝑎𝑛𝑠 +< 1 2 𝑘 𝑟 2 > =2𝑁<𝐾 > 𝑡𝑟𝑎𝑛𝑠 =3𝑁 𝑘 𝐵 𝑇 We will assume that ideal solids are incompressible 𝑑𝑈=𝑑𝑄 and 𝑐 𝑚𝑜𝑙 = 1 𝑛 𝑑𝑈 𝑑𝑇 =3 N A k B

Boltzmann Distributions and Quantum Oscillators “The best thing in the world”-V, 2018/4/7 For a Quantum Oscillator, energy is discretized by 𝜖=hf and 𝑈 𝑞 =𝑞𝜖= 𝑘 𝐵 𝑇=qhf Total number of states given N oscillators and q energy quanta Ω 𝑁,𝑞 = 𝑁−1+𝑞 ! 𝑞! 𝑁−1 ! 𝑆=𝑘 ln Ω Δ𝑆(𝑞→𝑞+1)=𝑘 ln 𝑞+𝑁 𝑞+1 ≈𝑘 𝑁 𝑞 1 𝑇 = d𝑆 d𝑈 ≈ 𝑘𝑁 𝑞ℎ𝑓 Probability of State having energy 𝐸 𝑛 : 𝑃 𝐸 𝑛 = 𝑒 − 𝐸 𝑛 𝑘 𝐵 𝑇 𝑍 Partition function: 𝑍= Σ 𝑎𝑙𝑙 𝑠𝑡𝑎𝑡𝑒𝑠 𝑒 − 𝐸 𝑛 𝑘 𝐵 𝑇 REMEMBER TO TAKE INTO ACCOUNT DEGENERACY!!! “Actually the Partition function is my favorite thing in the world”-V, 10 mins later R S

More 213 Stuff Atmospheres: Δ𝑈=𝑚𝑔ℎ therefore 𝑃 ℎ 𝑃 0 = 𝑒 − 𝑚𝑔ℎ 𝑘 𝐵 𝑇 and <ℎ> = 𝑘 𝐵 𝑇 𝑚𝑔 “You get this from the Boltzmann distribution, the best thing in the world”-V, 30mins later

Exam Advice When doing problems, PLEASE PLEASE PLEASE CHECK YOUR UNITS!!! You will be shocked how many questions you can solve just matching units and having an idea of what they are asking for Know when and how to use your equation sheet Don’t panic, just keep on moving Make sure you are in the right mindset going into the exam Spend your time showing what you know DON’T CHEAT

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