АЗ The microcanonical entropy of some gas is Sm Bin [V-Vo)Nf(E,N] , where V is the...
02) Consider a hydrostatic system represented by the thermodynamic variables volume V, pressure P and temperature T. a) Consider entropy S = S(T, V) and derive the equation TdS. Tas = Cvat +T (1) dV. V Show that this equation can be written as follows BT TdS = CydT + PDV where Cv is the thermal capacity at constant volume, B is the isobaric expansiveness and K is isothermal compressibility: b) Consider a gas described by the equation of state...
. (40 points) Consider an insulated container of volume V2. N idea gas molecules are initially confined within volume V, by a piston and the remaining volume V2 - Vi is in vacuum. Let T,, P1, E1, S, Al, Hi, G, be the temperature, pressure, energy, entropy, Helmholtz free energy, enthalpy, and Gibbs free energy of the ideal gas at this state, respectively V1 V2-V Using Sackur-Tetrode equation for the entropy of ideal gas where kB R/NA is Boltzmann's constant...
The Sackur-Tetrode Equation gives the entropy of a sample of n moles of monatomic ideal gas as a function of its internal energy U and volume V S(U, V) = 5/2 n R + n R In (V/n N_A(4piM U/3nN^2_Ah^2)^3/2) In the equation, R is the gas constant, M is the molar mass, N_4 is Avogadro's number, and h is Plank's constant. The equation can be derived using S = k ln W and directly computing W, the number of...
4. 10 points A monoatomic gas obeys the van der Waals equation: N²a P= NT V - Nb V2 where N is the number of particles and a and b are known constants and t = kbT. The gas has a heat capacity Cy = 3N/2 in the limit V +0. a) Using the thermodynamic identities and the equation of state prove that acv = 0. av т (3 pts) b) Use the result of part a) to determine the...
B.2 The multiplicity of a monatomic ideal gas is given by 2 = f(N)VN U3N/2, where V is the volume occupied by the gas, U its internal energy, N the number of particles in the gas and f(N) a complicated function of N. [2] (i) Show that the entropy S of this system is given by 3 S = Nkg In V + ŽNkg In U + g(N), where g(N) is some function of N. (ii) Define the temperature T...
3. A gas obeys the equation of state PV = nRT - an'/V, where n is the number of moles of gas and a is constant. Substitute with rearrangement into the differential equation for work, and integrate from Vito V2 to find an equation for the work done by this gas as the result of a reversible isothermal process. Show algebraically that the work is proportional to n to the first power.
3. A gas obeys the equation of state PV = nRT - an'/V, where n is the number of moles of gas and a is constant. Substitute with rearrangement into the differential equation for work, and integrate from Vito V2 to find an equation for the work done by this gas as the result of a reversible isothermal process. Show algebraically that the work is proportional to n to the first power.
1. Show that for a classical ideal gas, Q1 alnQ1 NK Hint: Start with the partition function for the classical ideal gas ( Q1) and use above equation to find the value of right-hand side and compare with the value of r we derive in the class. (Recall entropy you derived for classical gas) NK Making use of the fact that the Helmholtz free energy A (N, V, T) of a thermodynamic system is an extensive property of the system....
(Thermodynamics, Maxwell relations. Mathematics) 3 Consider the microcanonical formula for the equilibrium energy E(S, V,N) of some general system.58 One knows that the second deriva- tives of E are symmetric; at fixed N, we get the same answer whichever order we take partial derivatives with respect to S and V (a) Use this to show the Marwell relation (3.70) OS S,N (This should take two lines of calculus or less.) Generate two other similar formulae by taking other second partial...
(a) One mole of a monoatomic van der Waals gas obeys the equation of state A3. ) (V-b)=RT (p+ and its internal energy is expressed as U CvT where Cv is the molar isochoric heat capacity of an ideal gas. The gas is initially at pressure p and volume V (i) Explain the physical meaning of the parameters a and b in the equation of state of the gas (ii) Write down the equation that defines entropy in thermodynamics. Define...