Tarea7% - CINVESTAVjmendez/tarea7.pdf · Tarea7%! 1....
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Tarea 7 1. (Phillies) Derive (17.20)(17.22) for ! , ! and ! in terms of the ( ). 2. (Phillies) Compute ! in terms of the ( ). 3. (Phillies) Confirm the steps leading from (17.30) to (17.32). 4. (Phillies) Derive (17.33) from your expression for ! in terms of the ( ). 5. (Phillies) Consider the “sticky hard sphere”, whose potential energy is = +∞ for < ! for ≤ < 0 for ≥ Evaluate ! . Compute and Ξ for this system in the approximation that the higher virial coefficients ( ! for ≥ 3) are negligible. 6. (McQuarrie) Show that ! = − 1 6 ! () ! ! !!(!)/! ! ! 4 ! is equivalent to ! = − 1 2 !!" ! − 1 ! ! 4 ! . State the condition on () that is necessary. 7. (McQuarrie) Derive la ecuación de la energía ! = 3 2 + 2 ! () 4 ! ! ! . 8. (McQuarrie) Derive la ecuación de la presión ! = − ! 6 ! 4 ! ! ! . 9. (Hansen y McDonald) Demuestre que ! = ( − ! ) ! !!! . 10. (Hansen y McDonald) Demuestre que ! , ′ = ( − ! )(′ − ! ) ! !,!!! !!! . 11. (Hansen y McDonald) Demuestre que () ≡ 1 (′ + )(′) ! = + . 12. (Hansen y McDonald) Demuestre que ≡ 1 − = 1 + 2 ! + ℎ , donde ℎ () es la transformada de Fourier de ℎ = − 1.
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Transcript of Tarea7% - CINVESTAVjmendez/tarea7.pdf · Tarea7%! 1....
Tarea 7 1. (Phillies) Derive (17.20)-‐(17.22) for 𝐵!, 𝐵! and 𝐵!in terms of the 𝑍(𝑖). 2. (Phillies) Compute 𝐵! in terms of the 𝑍(𝑖). 3. (Phillies) Confirm the steps leading from (17.30) to (17.32). 4. (Phillies) Derive (17.33) from your expression for 𝐵! in terms of the 𝑍(𝑖). 5. (Phillies) Consider the “sticky hard sphere”, whose potential energy is
𝑉 𝑟 =+∞ for 𝑟 < 𝑎 𝑉! for 𝑎 ≤ 𝑟 < 𝑏0 for 𝑟 ≥ 𝑏
Evaluate 𝐵!. Compute 𝑃 and Ξ for this system in the approximation that the higher virial coefficients (𝐵! for 𝑗 ≥ 3) are negligible.
6. (McQuarrie) Show that
𝐵! = −1
6𝑘!𝑇𝑟𝑑𝑢(𝑟)𝑑𝑟
!
!𝑒!!(!)/!!!4𝜋𝑟!𝑑𝑟
is equivalent to
𝐵! = −12 𝑒!!" ! − 1
!
!4𝜋𝑟!𝑑𝑟.
State the condition on 𝑢(𝑟) that is necessary.
7. (McQuarrie) Derive la ecuación de la energía 𝐸
𝑁𝑘!𝑇=32+
𝜌2𝑘!𝑇
𝑢(𝑟)𝑔 𝑟 4𝜋𝑟!!
!𝑑𝑟.
8. (McQuarrie) Derive la ecuación de la presión
𝑃𝑘!𝑇
= 𝜌 −𝜌!
6𝑘!𝑇𝑟𝑑𝑢 𝑟𝑑𝑟 𝑔 𝑟 4𝜋𝑟!
!
!𝑑𝑟.
9. (Hansen y McDonald) Demuestre que
𝜌 ! 𝒓 = 𝛿(𝒓− 𝒓!)!
!!!.
10. (Hansen y McDonald) Demuestre que
𝜌 ! 𝒓, 𝒓′ = 𝛿(𝒓− 𝒓!)𝛿(𝒓′− 𝒓!)!
!,!!!!!!
.
11. (Hansen y McDonald) Demuestre que
𝐺(𝒓) ≡1𝑁 𝜌(𝒓′+ 𝒓)𝜌(𝒓′) 𝑑𝒓! = 𝜌𝑔 𝒓 + 𝛿 𝒓 .
12. (Hansen y McDonald) Demuestre que
𝑆 𝒌 ≡1𝑁 𝜌 𝒌 𝜌 −𝒌 = 1+ 2𝜋 !𝜌𝛿 𝒌 + 𝜌ℎ 𝒌 ,
donde ℎ(𝒌) es la transformada de Fourier de ℎ 𝒓 = 𝑔 𝒓 − 1.
