2. Consider free expansion of a gas when the internal energy U remains constant. Derive: a) the e...
2. (25 pts) Derive the (a) Maxwell relation for the Helmholtz Free Energy F=U-TS. Show ALL steps and justifications in your derivation. Using your result in (a) comment on how (b) the entropy behaves for an isothermal expansion of an ideal gas. Finally, show the validity of the following equations (c) U = F-TOOF) -T2 and at (T) 01 (d) C =-1(
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...
Derive) A) the work done by n moles an ideal gas at temperature T in an isothermal expansion from V1 to V2 B) The entropy change of n moles of an ideal gas at Temperature T undergoing an isothermal expansion from V1 to V2
Er<E EF E E> E) W>0) and Polytropic PathsSpcl Cases for Ideal Gas n 0 constant pressure n 1 constant temperature n k constant entropy, adiabatic (q 0) n constant volume and W<0 W 0 For air R 0.287 kJ/kg-K and k Cp/Cv 1.4 if pi 300 kPa, v, 0.861 m3/kg then T, 900 K For T Ta if pa /p 3 then va = m2/kg pv RT and (T, n/(n-1) p2 V1 For vi v if po /p1 3...
(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...
9. In a free expansion of a perfect gas (also called Joule expansion), we know that internal energy U does not change, and no work is done. However, the entropy must increase because the process is irreversible. Are these statements compatible with the first law of thermodynamics
The internal energy, U, is the total energy of a system. For any isolated system, the internal energy is constant. U is a state function, meaning that any path used in calculating AU ill result in the same answer. For any pure substance or fixed mixture of substances, the internal energy, U, can be determined from any two of the variablesP, V, and T. It is often most convenient to choose V and T as the variables. It is helpful,...
. (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...
A monatomic ideal gas is initially at volume, pressure, temperature (Vi, Pi, Ti). Consider two different paths for expansion. Path 1: The gas expands quasistatically and isothermally to (Va, Pz. T2) Path 2: First the gas expands quasistatically and adiabatically (V2, P.,T-),where you will calculate P T. Then the gas is heated quasistically at constant volume to (Va. P2 T1). a. Sketch both paths on a P-V diagram. b. Calculate the entropy change of the system along all three segments...
4. [After Reif Problem 5.1] When an ideal gas undergoes an adiabatic (thermally insu- lated) quasi-static expansion, its pressure and volume are related by p = constant. where γ = cp/cv is the ratio of heat capacities. If the gas expands from an initial volume Vi at temperature T to a final volume V2, calculate the final temperature T2 in terms of γ, Vi, Ti, and ½.