The multiplicity for an ideal gas is given in your book, sigma(U,V,N)= f(N)V^NU^
ID: 2254938 • Letter: T
Question
The multiplicity for an ideal gas is given in your book, sigma(U,V,N)=f(N)V^NU^-3N/2. Consider a related system in which f(N) = 1 (with correct units) and so the multiplicity will be given by (sigma)=V^N U^3N/2 .
Two such isolated systems are in contact with each other and can exchange energy and volume but not particles. System A has 1600 particles and system B has 2300 particles and the two systems share a total energy of 980 and total volume of 730 (in the correct units).
Using a computer, plot the total entropy of the two systems as a function of volume in A; keep the energy in A and B fixed at some value of your choosing. Also, plot the total entropy of the two systems as a function of energy in A; keep the volume in A and B fixed at some value of your choosing.
When the system reaches equilibrium what is the volume and energy of system A?
(FULL SOLUTION PLEASE)
(PICTURE OF INFO PUT INTO MATLAB OR MATHEMATICA)
THANK YOU
Explanation / Answer
Isothermal process :
A process during which energy enters or leaves a system in such a way that that a constant temperature is maintained. Astrophysical examples include the descent of a protostar down the Hayashi track and the collapse of an evolved star to become a white dwarf. Compare with adiabatic
An isothermal process is one that is carried out at constant temperature. This can be accomplished by placing a system in contact with a large heat reservoir and allowing heat to be transferred between the system and the reservoir. For an ideal gas, the energy depends only on temperature through
3
E=------ nRT
2
In an isothermal process, therefore, all of the heat absorbed by the system is converted into work performed in an expansion. The change in internal energy is 0:
del. E =3/2 nR del.T = 0
q= -W
since Del.T =0 in an isothermal process for an ideal gas.
If the work were performed at constant pressure, then it could be computed from
W= -Pext. del.V
However, in a general isothermal process, the pressure is NOT held constant. Thus, in order to compute the work performed in an isothermal expansion, we need to know how the pressure depends on the volume. For an ideal gas, this dependence is specified via the ideal gas law:
P=nRT/V = P(V)
However, if we want to know how much work the gas does in expanding from a volumeV1 to a volume V2 , we cannot use a simple formula like W = -P del.V = -P (V2-V1) since the pressure varies over the range of volumes. Rather, we need to compute the work done over infinitesimally small volume changes dV and evaluate the pressure at each new volume reached. That is, we need to integrate over the volume range, using the functional dependence of P on V:
dW = -P (V) .dV
W =
Since q=-W the heat absorbed will be given by
q = nRT ln (V2/V1)
Thermal Equilibrium :
It is an stage in the reaction at which there is neither icrease or decrease in the thermal content of the reaction.