1 Quadrature Debye's formula for the heat capacity of a solid is Cv = 9g(u) kN, (1) where the function g(u) is defined as p1/u 1¹e² 2³ Sol (e² - 17² d g(u) = y³ (2) 1)2 0 and where the terms in this equation are: N = number of particles (atoms) in the solid, k = Boltzmann constant, k = 1.38-10-23 m² kg s-2 K-¹ u = T/OD, (dimensionless temperature), T = absolute temperature in kelvin, OD = Debye temperature, which is a property specific to the solid. (a) Compute just g(u) using Eqn. 2, from u = 0 to 1.0 in intervals of 0.05 and plot the results. You should get something like the curve given in https://en.wikipedia. org/wiki/Debye_model#/media/File:DebyeVSEinstein.jpg. What happens when u is exactly zero? How big does u need to be before we do not hit a singularity? (b) Now find the isochoric heat capacity, Cu, of iron, Fe, at a temperature of T = 300K in appropriate units. Hint: You will find the Debye temperatures, Op, for various solids listed at https: //en.wikipedia.org/wiki/Debye_model. You will also need the density of iron, and the molecular weight in order to find N which is the number of Fe atoms in 1 cm³. These are easy to find in tables or on the internet. You can test your algorithm on the fact that C 3.537 J/cm³/K. Value from https://en.wikipedia.org/wiki/Table_of_specific_heat_capacities. dz,
1 Quadrature Debye's formula for the heat capacity of a solid is Cv = 9g(u) kN, (1) where the function g(u) is defined as p1/u 1¹e² 2³ Sol (e² - 17² d g(u) = y³ (2) 1)2 0 and where the terms in this equation are: N = number of particles (atoms) in the solid, k = Boltzmann constant, k = 1.38-10-23 m² kg s-2 K-¹ u = T/OD, (dimensionless temperature), T = absolute temperature in kelvin, OD = Debye temperature, which is a property specific to the solid. (a) Compute just g(u) using Eqn. 2, from u = 0 to 1.0 in intervals of 0.05 and plot the results. You should get something like the curve given in https://en.wikipedia. org/wiki/Debye_model#/media/File:DebyeVSEinstein.jpg. What happens when u is exactly zero? How big does u need to be before we do not hit a singularity? (b) Now find the isochoric heat capacity, Cu, of iron, Fe, at a temperature of T = 300K in appropriate units. Hint: You will find the Debye temperatures, Op, for various solids listed at https: //en.wikipedia.org/wiki/Debye_model. You will also need the density of iron, and the molecular weight in order to find N which is the number of Fe atoms in 1 cm³. These are easy to find in tables or on the internet. You can test your algorithm on the fact that C 3.537 J/cm³/K. Value from https://en.wikipedia.org/wiki/Table_of_specific_heat_capacities. dz,
Computer Networking: A Top-Down Approach (7th Edition)
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Using python or matlab to compute this.
Transcribed Image Text:1 Quadrature
Debye's formula for the heat capacity of a solid is
Cv = 9g(u) kN,
(1)
where the function g(u) is defined as
p1/u 1¹e²
2³ Sol (e² - 17² d
g(u) = y³
(2)
1)2
0
and where the terms in this equation are:
N = number of particles (atoms) in the solid,
k = Boltzmann constant, k = 1.38-10-23 m² kg s-2 K-¹
u = T/OD, (dimensionless temperature),
T = absolute temperature in kelvin,
OD = Debye temperature, which is a property specific to the solid.
(a) Compute just g(u) using Eqn. 2, from u = 0 to 1.0 in intervals of 0.05 and plot the
results. You should get something like the curve given in https://en.wikipedia.
org/wiki/Debye_model#/media/File:DebyeVSEinstein.jpg.
What happens when u is exactly zero? How big does u need to be before we do not
hit a singularity?
(b) Now find the isochoric heat capacity, Cu, of iron, Fe, at a temperature of T = 300K
in appropriate units.
Hint: You will find the Debye temperatures, Op, for various solids listed at https:
//en.wikipedia.org/wiki/Debye_model. You will also need the density of iron,
and the molecular weight in order to find N which is the number of Fe atoms in 1
cm³. These are easy to find in tables or on the internet.
You can test your algorithm on the fact that C 3.537 J/cm³/K. Value from
https://en.wikipedia.org/wiki/Table_of_specific_heat_capacities.
dz,
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