(2) Consider a fictitious system, whose Helmholtz free energy is expressed asF(T,V,N) NRT 1 - α--(1-ax).where T,V,N are temperature, volume, number of particles, respectively. R, a are positiveconstants. Note that the equation of state for ideal gas does not apply to this system.(a) Calculate the pressure.(b) Obtain Gibbs free energy as a function of (T,p,N) by using pressure p and Legendretransform: F(T,V,N) → G(T,p,N).OGдр(c) From the definition of Gibbs free energy, what thermodynamic quantity should give?Does G(T,p,N) above give this quantity?(d) The fictitious Helmholtz free energy in this problem has an undesirable property. Explainwhy this F is not expected to exist in reality.

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Answered: (2) Consider a fictitious system, whose…

University Physics Volume 2

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Chapter2: The Kinetic Theory Of Gases

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Problem 68P: Using the method of the preceding problem, estimate the fraction of nitric oxide (NO) molecules at a...

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One process for decaffeinating coffee uses carbon dioxide ( M=44.0 g/mol) at a molar density of about 14,0 mol/m3 and a temperature of about 60 . (a) Is CO2 a solid, liquid, gas, or supercritical fluid under those conditions? (b) The van der Waals constants for carbon dioxide are a=0.3658 Pa m6/mol2 and b=4.286105 m3/mol. Using the van der Waals equation, estimate pressure of CO2 at that temperature and density. `
define the chemical potential in terms of derivatives of the ĈE energy E and enthalpy H. For a one component system, these are u= ON and Evaluate these expressions for an ideal gas and compare to ON µ = -kT In (kT/PA³ ) from H=| P,s OF and |= 1 TP ON v.r
Problem 1: In statistical mechanics, the internal energy of an ideal gas is given by: N. aNkB 2/3 (3NKB U = U(S,V) = е where a is a constant. 1- Show that the variation of the internal energy is given by: 2 dS - \3V 2 dU = dV \3NkB 2- Using the fundamental relation of thermodynamic dU = T.ds – p. dV, show that the equation of state PV = nRT follows from the first expression of U.
When a liquid is compressed its Gibbs free energy is increased.To a first approximation the increase in the molar Gibbs free energy can be calculated by using (∂ ̄G/∂P)T= ̄V, assuming a constant molar volume.What is the change in molar Gibbs free energy for liquid water when it iscompressed to 1000 bar?
Give the temperature T of 1 mole of ideal gas as a function of the pressure P, volume V, and the gas constant R and give the internal energy U of a rigid diatomic ideal gas as a function of its temperature T and the gas constant R.
Show that for an ideal gas (@T/ƏV)u = 0, and (@T/ƏP)H= 0.
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Calculus can be used to derive a relation between the critical constants and the van der Waals parameters of a gas. The critical point of a van der Waals gas occurs where the isotherm has a flat inflexion, which is where dp/dVm = 0 (zero slope) and d2p/dVm2 = 0 (zero curvature). (a) Derive expressionsfor these first and second derivatives by differentiating the van derWaals equation of state. Hence find expressions for the critical constants in terms of the van derWaals parameters. (b) Show that the value of the compression factor at the critical point is (3)/(8) . (c) The values of the van der Waals parameters of nitrogen dioxide, N02, are a = 5.354 dm6 bar mol-1 and b= 0.04424 dm3 mol-1. Calculate the values of the criticalconstants and confirm that the compression factor does indeedhave a value of (3)/(8) at the critical point.
1.7 Ideal gas response functions Find the thermal expansion coefficient and the isothermal compressibility for an ideal gas and show that in this case Cp - Cy = - a² can be reduced to Cp – Cy = Nkg for the molar specific heats. TV Кт
Consider 1 mole of a van der Waals gas. (i) Derive the expressions for the pressure, pc, temperature, Tc, and volume, Vc, in the critical point of a van der Waals gas in terms of parameters a, b and R. Derive the vdw equation in reduced coordinates p =,= 7, V = V/ (ii) (iii) Find how many times the gas temperature exceeds its critical temperature if the gas pressure is 4 times as high as critical pressure and the volume of gas is equal to twice the critical volume.
Consider the earth's atmosphere as an ideal gas of molecular weignt u in a uniform gravitational field. Let g denotc the acceleration due to gravity. (a) If z denotes the height above sea level, show that the change of atmos- pheric pressure p with height is given by dp where T is the absolute temperature at the height z. (6) If the decrease of pressure in (a) is due to an adiabatic expansion, show that dp Y dT Y -iT (c) From (a) and (b) calculate dT/dz in degrees per kilometer. Assume the atmosphere to consist mostly of nitrogen (N) gas for which y - 1.4. (d) In an isothermal atmosphere at temperature T, express the pressure p at height z in terms of the pressure po at sea level. (e) If the sea-level pressure and temperature are po and To, respectively, and the atmosphere is regarded as adiabatic as in part (b), find again the pressure p at height z.
In this problem you will investigate the behavior of a van derWaals fluid near the critical point. It is easiest to work in terms of reduced variables throughout. Still working in the same limit, find an expression for the difference in volume between the gas and liquid phases at the vapor pressure. You should find (Vg -Vi) ex: (Tc -Tl, where (3 is known as a critical exponent. Experiments show that (3 has a universal value of about 1/3, but the van der Waals model predicts a larger value.

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