For a dilute gas of N monatomic particles with mass m and total energy E, use the Sackur- Tetrode equation for the entropy S V = log + NkB to derive expressions for the pressure and internal energy in terms of the temperature T and volume V. [You may use that X₁ = 3πh² N/(mE).] th
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![For a dilute gas of N monatomic particles with mass m and total energy E, use the Sackur-
Tetrode equation for the entropy
S
V
=
log
+
NkB
to derive expressions for the pressure and internal energy in terms of the temperature T and
volume V.
[You may use that X₁ = 3πh² N/(mE).]
th](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F3b765a29-fcb0-4895-9cef-5ef60c256f5b%2Ff1247c9d-3b74-4675-a47a-fbd40333bc78%2Ftbl0vwg_processed.png&w=3840&q=75)
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- 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.Write the mathematical relationships for each of the following Conclude the thermodynamic properties (parameters) of ideal gas same as ( energy , prussure, entropy , specific heat at fixed prussure , specific heat at fixed volume ) e 12:31 Va. Find an appropriate expression for the change in entropy in the following two cases: 1) S=S(T, V) 2) s= S(T, P) Where: S is entropy, T is temperature, V is volume, P is pressure b. Prove the following two themodynamie property relationships (똥),-() 8C, Where: T, P. V are temperature, pressure and volume, respectively. C, and C, are specific heats at constant volume and constant pressure, respectively.
- 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.rProblem 3: Starting with the expression derived in the lecture notes for the multiplicity of an ideal 1 n3N/2 (2m)³N/2 N! (3N/2)! h3N gas VNU3N/2 derive the Sackur-Tetrode expression for the entropy.Problem #2 For heat exchange between a thermal reservoir at 300 K and a constant volume system containing one mole of monatomic ideal gas: a) Derive the equation for the total change in entropy for a designed initial system temperature Tj. b) Plot AStotal vs. Tsys for the initial system temperature ranging from 160 K to 500 K in increments of 10 K (i.e., Tsys = 160 K, 170 K, ... , 500 K). Use Matlab, Excel or similar plotting software for your plot. Label the plot axes and include units. %D
- Entropy (s) is aun extensive variable . & S(U,V,N) Uo Internal Energy V→ Volume Number of particles find an Euler equation For entropeUsing the graph below and by considering the fundamental equation of thermodynamics, justify the differences in behaviours of Gibbs as a function of pressure for gaseous, liquid and solid forms of a substance. Gas Liquid Solid Pressure, p The molar volume of liquid water is 18.05 cm³. By considering the fundamental equation of chemical thermodynamics, calculate the change in molar Gibbs energy, dG, when the pressure is increased by 0.1 bar. Gibbs energy, GCalculate the number of microstates that are available in a single atom of carbon in graphite.The standard molar entropy (S°) of carbon is 5.7 J/(mol · K) and the Boltzmann constant (kB) is1.381 × 10 ―23?/?. HELP PLEASE
- dU =T dS – P dV Re-arrange the thermodynamic identity by solving for dS. Assume we are applying this new expression to an ideal gas, re-write the du term in terms of the appropriate heat capacity and temperature. Re-write the coefficient of the dV term in terms of volume. Connect these “classical" expressions for entropy to the "modern" interpretation where we are concerned with the uncertainty associated with position and momentum of the particles in an ideal gas. Answer with sentences. Now, re-write the expression in terms of enthalpy so you can connect entropy with enthalpy. Expalin why this final expression makes sense.a) Find the difference of molar-specific heats Cp - C, for the gas covered by the equation of state: (P+) (V - b) = RT b) Determine the volume of the hypersphere in N-dimensional space. c) An ideal gas consists of N monatomic molecules. Using the grand canonical ensemble, (T-μ) distribution, calculate the chemical potential, pressure, and entropy.Problem 3: Consider an Einstein solid with N oscillators and total energy U = qe, in the limit N,q » 1 (with no assumptions made about the relative size of N and q). + N° (9 +N\9 a) Starting with this formula, find an expression for the entropy of an Einstein solid as a function of N and q. Explain why factors omitted from the formula have no effect on the entropy. b) Derive an expression for the temperature of the solid, as a function of N and q. Simplify your expression as a much as possible. c) Invert the result of part (c) to get the energy U as a function of temperature T. As always, simplify the final result as much as possible. d) Show that, in the high temperature limit (q » N), the heat capacity is C = Nkg. (Hint: when x is small, e* = 1+ x.) Is this the result you would expect? Explain. e) Plot energy U vs. temperature T using dimensionless variables, Cy/(Nkg) vs. t = kgT/e, for t in the range from 0 to 2. Discuss your prediction for the heat capacity at low temperature…
