Starting with the Clausius Inequality, ∂S ≥ ∂q/T, can you prove that, under conditions of constant pressure and entropy, for the total entropy to increase, ∂H ≤ 0 J?
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Starting with the Clausius Inequality, ∂S ≥ ∂q/T, can you prove that, under conditions of constant pressure and entropy, for the total entropy to increase, ∂H ≤ 0 J?
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- Prove that the entropy of mixing of an ideal mixture has aninfinite slope, when plotted vs. x, at x = 0 and x = l.A highly non-ideal gas has an entropy given by S=aNU/V, where the internal energy, U is a function of T.Find the pressure, the expression for the heat capacity at constant volume, and the chemical potential.A mass m of water is heated reversibly from temperature T to T, at a constant pressure of P. In this problem, we are going to determine an expression for the change in entropy, AS. Assume we can heat the given water infinitesimally slowly so that the process is reversible. Therefore, heat in any infinitesimal step is given by the following: dQ = mc dT, where c is the specific heat and is constant. Calculate the change in entropy in cal/K for a sample of water with mass m= 1.6 kg and changing temperature from T1 = 24.2°C to T2= (24.2+10)°C. The specific heat c of water is 1,000 cal/kg/K. AS=
- For one component gas that is confined in a box with volume V. We can get the entropy of the gas as S= Nk, in- where N is the total a² number of atoms, a is the radius of the atom. Can you guess how it is obtained?For either a monatomic ideal gas or a high-temperature Einstein solid, the entropy is given by Nk times some logarithm. The logarithm is never large, so if all you want is an order-of-magnitude estimate, you can neglect it and just say S - Nk. That is, the entropy in fundamental units is of the order of the rv number of particles in the system. This conclusion turns out to be true for most systems (with some important exceptions at low temperatures where the particles are behaving in an orderly way). So just for fun, make a very rough estimate of the entropy of each of the following: this book (a kilogram of carbon compounds); a moose (400 kg of water); the sun (2 x 1030 kg of ionized hydrogen).(a) What is the entropy of an Einstein solid with 4 atoms and an energy of 18ε? Express your answer as a multiple of kB . The entropy of the solid is ____ kB.(b) What is the entropy of an Einstein solid in a macropartition that contains 9 ×10690×10690 microstates? Express your answer as a multiple of kB. The entropy of the solid is ____ kB.
- 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…Plot the function dS/dT for a two-level system, the temperature coefficient of its entropy, against kT/ε. Is there a temperature at which this coefficient passes through a maximum? If you find a maximum, explain its physical origins.The partition function of a hypothetical system is given by In Z = «TªV where a is a constant. Evaluate the mean energy E, the pressure P, and the entropy S.
- For one component gas that is confined in a box with volume V. V We can get the entropy of the gas as S = Nk, In where N is the total number of atoms, a is the radius of the atom. Can you guess (or work out) how it is obtained?A mass m of water is heated reversibly from temperature T1 to T2 at a constant pressure of P. In this problem, we are going to determine an expression for the change in entropy, ΔS.Assume we can heat the given water infinitesimally slowly so that the process is reversible. Therefore, heat in any infinitesimal step is given by the following: dQ = mc dT, where c is the specific heat and is constant. Part (a) Write an expression for the change in entropy ΔS for the system. Part (b) Calculate the change in entropy in cal/K for a sample of water with mass m = 1.9 kg and changing temperature from T1 = 22.9°C to T2 = (22.9+10)°C. The specific heat c of water is 1,000 cal/kg/K.Hi, could I get some help with this macro-connection physics problem involving isothermal expansion? The set up is: For an isothermal reversible expansion of two moles of an ideal gas, what is the entropy change of the a) gas and b) the surroundings in J/K to 4 digits of precision if the gas volume quadruples, assuming NA = 6.022e23 and kB = 1.38e-23 J/K? Thank you.