Starting with Ec 7-12
(1)All we have to do is to compute the classical configurational integral.
(2)where A is the area of the cylinder
Now, we know that
then
(5)Because
(6)Finally
(8)Starting with Ec 7-12
(1)All we have to do is to compute the classical configurational integral.
(2)where A is the area of the cylinder
Now, we know that
then
(5)Because
(6)Finally
(8)Teniendo la funcion de partición de un cristal, la cual tiene al forma:
(1)Donde $hv/k=\Theta_E$ es una constante característica de un cristal, y $U_o$es la energía de sublimación de un cristal. Calcular el calor específico y muestre que a altas temperaturas uno obtiene la ley Dulong y Petit, esto es $C_v \rightarrow 3Nk$.
Primero, recordemos el calor específico tiene la forma:
(2)Y sabemos que la expresión de la energia promedio tiene la forma:
(3)¨Por lo que, del logaritmo de Q, el cual es:
(4)Y la derivada de esto toma la forma:
(6)Multiplicando por $T^2k$ tenemos:
(7)Por lo que, derivando la expresión anterior con respecto a T
(8)Y al hacer el límite de esta expresión a $T \rightarrow \infty$ se obtiene
(11)Teniendo la función de partición de un cristal, la cual tiene al forma:
(1)donde
(2)Donde I es el momento de inercia de la molécula; v es su frecuencia vibracional fundamental; y $\mu$ es su momento dipolar. Usando la función de partición y utilizando la relación termodinámica,
(3)donde $M=N \bar{\mu}$, donde $\bar{\mu}$ es el momento dipolar promedio de una molécula en la dirección del campo externo $\xi$, muestre que
(4)Grafique los resultados desde $\xi =0$ hasta $\xi= \infty
Primero obtengamos dA, el cual tomará la forma:
(5)Pero solo nos interesa igualar el término $\dfrac{\partial A}{\partial \xi}d \xi$, pues al igualarlo con la expresión dada del diferencial de dA, obtendremos que
(6)Por lo que
(7)Ahora A, tiene la forma:
(8)Por lo que su derivada con respecto a $\xi$ tendrá la forma
(11)Ahora, como es $Ln [q]$ , y solo sobrevivirán aquellos términos dependientes de $\xi$, los cuales son
(12)Entonces tendremos la expresión
(13)Multiplicando por -1/N para obtener
(15)And looking at the graph, we can tell $\xi \rightarrow \infty$, $\bar{\mu} \rightarrow \mu$
determine the various thermodynamics properties of an Einstein crystal
i wonder if this site is operating or not
Hello all,
Could anyone help me in proving Eq. (21-292)? It seems that is quite simple but I cannot understand why the ensemble average in the right-hand side of Eq. (21-297) reduces to Eq. (21-298).
Regards
MN
You have a distribution
(1)with the constraints
(2)Now, you have the logarithm of your distribution (using Stirling's approximation in the second equality)
(3)Then, the Lagrangian of optimization is
(4)Taking the derivative with respect to $a_j$
(5)Rearranging you have
(6)Taking the exponential and summing over $j$:
(7)where $e^{-\alpha}=\left(\sum_j\Omega_je^{-\beta E_j}\right)^{-1}$ and $\beta=1/k_BT$. Finally you have the answer:
(8)—-
Please help me with this problem!
All I see here is "unsupported math environment ,tabular'" can someone repost the solution?
Results above using the one particle partition function, should be replaced with the full partition function
(1)Will result in a factor of N multiplying P, E, and C.
Entropy is calculated using:
(2)Given
(1)(1) Find the density of states and then use it to find the q_trans:
We can write:
(2)We assume that the number of states ($\Phi(\epsilon)$) with energy $\epsilon$ can be approximated by 1/4 the area of a circle with radius R. Then we have
(3)Then the number of states between $\epsilon$ and $\epsilon +d\epsilon$ is given by:
(4)Now using:
(5)gives:
(6)(2) Find $q_{trans}$ using another method:
This time we will sum over states instead of summing over levels, we write:
(7)Evaluating the integral gives:
(8)Which is fortunately the same we obtained by summing over levels.
(3) Find heat capacity, U, S, and EOS
To find the EOS we use:
(9)To find E:
(10)Finally to find $C_A$:
(11)Yes, What part are you having difficulty with specially? Also, I believe the answers will be posted on Monday to HW2.
Can anyone solve this problem?
Can anyone solve this problem?Thanks a lot!
can anyone please solve Q 16 of chapter 12?
This is almost the same than 2-17. You need to use the same equations to solve it, but here it goes:
To get the pressure, we use (sorry for the notation, but I really don't know how to write the solution in a better way. You can copy-paste in LaTeX…)
\bar{P} = kT (\frac{\partial ln Q}{\partial V})_N
with Q(N,V,T) = \frac{1}{N!} (frac{2 \pi m k T}{h^2})^{3N/2} V^N.
Doing the derivatives we get
\bar{P} = \frac{kTN}{V}.
In order to get the energy, we use
\bar{E} = kT^2 (\frac{\partial ln Q}{\partial T})_N
and the same Q, of course. So the energy is
\bar{E} = \frac{3 kTN}{2}
I have tried problem 2-14 but cant solve it. can anyone help?
State and use Euler's therom to show
(1)From equation (1-56)
(3)In The Grand Canonical Ensemble
(4)Thus we have
(5)From equation (3-12)
(7)We can rewrite (1) in term of Entropy in equation (3-12)
(8)Apply Euler's theorem
(11)Therefor
(12)We can rewrite it as
(13)And we can use the same method to state that p=kT(\frac{\partial\ln\Xi}{\partial V})
(14)