P3C.9 At low temperatures there are two contributions to the heat capacity of a metal, one associated with lattice vibrations, which is well-approximated by the Debye T-law, and one due to the valence electrons. The latter is linear in the temperature. Overall, the heat capacity can be written (a) Assuming that the expression given above for the heat capacity applies, explain why a plot of C,m(T)/T against T² is expected to be a straight line with slope a and intercept b. (b) Use such a plot to determine the values of the constants a and b. (c) Derive an expression for the molar entropy at temperature T. (Hint: you will need to integrate Cm(T)/T.) (d) Hence determine the molar entropy of potassium at 2.0 K. Debye electronic Cp.m (T)= aT³ + bT The molar heat capacity of potassium metal has been measured at very low temperatures to give the following data T/K 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.437 0.560 0.693 0.838 0.996 1.170 1.361 1.572 (JK' mol')

P3C.9 At low temperatures there are two contributions to the heat capacity of a metal, one associated with lattice vibrations, which is well-approximated by the Debye T-law, and one due to the valence electrons. The latter is linear in the temperature. Overall, the heat capacity can be written (a) Assuming that the expression given above for the heat capacity applies, explain why a plot of C,m(T)/T against T² is expected to be a straight line with slope a and intercept b. (b) Use such a plot to determine the values of the constants a and b. (c) Derive an expression for the molar entropy at temperature T. (Hint: you will need to integrate Cm(T)/T.) (d) Hence determine the molar entropy of potassium at 2.0 K. Debye electronic Cp.m (T)= aT³ + bT The molar heat capacity of potassium metal has been measured at very low temperatures to give the following data T/K 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.437 0.560 0.693 0.838 0.996 1.170 1.361 1.572 (JK' mol')

Chemistry

10th Edition

ISBN:9781305957404

Author:Steven S. Zumdahl, Susan A. Zumdahl, Donald J. DeCoste

Publisher:Steven S. Zumdahl, Susan A. Zumdahl, Donald J. DeCoste

Chapter1: Chemical Foundations

Section: Chapter Questions

Problem 1RQ: Define and explain the differences between the following terms. a. law and theory b. theory and...

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For an ideal gas if the specific internal energy at a specific pressure and temperature of 20 C is u=339.6 kJ/kg, what is the specific internal energy if the pressure is doubled while the temperature stays the same.
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Suppose that attractions are the dominant interaction between gas molecules, and the equation of state is p = nRT/V – n2a/V2. Determine the work (W(non-ideal gas)) of reversible, isothermal expansion of such a gas from initial volume V (initial) = 20.0 L to final volume V(final) = 40.0 L if n = 2.00 mol, T = 300 K, and a = 3.621 atm-L2/mol2. Watch your units. Determine the work (W(ideal gas) of reversible, isothermal expansion of an ideal gas from initial volume V (initial) = 20.0 L to final volume V(final) = 40.0 L if n = 2.00 mol and T = 300 K. Show the difference W(non-ideal) – W(ideal). If all your calculations are done correctly, this result shows you the effect of attractive interaction between gas particles on the work done by the system.
gas undergoes an adiabatic expansion from 10 − 3 m 3 to 8 × 10 − 3 m 3 . The adiabatic line for the gas is given by P 3 V 5 = C (C is a constant). (a) Calculate the reversible work performed along the adiabatic line if the initial pressure is 10 5 P a and the initial volume = 10 − 3 m 3 . (b) What is the value of the constant C ? What is the unit of C ? (c) The gas reaches the final state in a two step process. An expansion at a constant initial pressure to the final volume followed by a decrease in pressure to the final volume 8 × 10 − 3 m 3 . What is the reversible work and heat in this process?
(a) Suppose that attractions are the dominant interaction between gas molecules, and the equation of state is p = nRT/V – n2a/V2. Determine the work (W(non-ideal gas)) of reversible, isothermal expansion of such a gas from initial volume V (initial) = 20.0 L to final volume V(final) = 40.0 L if n = 2.00 mol, T = 300 K, and a = 3.621 atm-L2/mol2. Watch your units. (b)Determine the work (W(ideal gas) of reversible, isothermal expansion of an ideal gas from initial volume V (initial) = 20.0 L to final volume V(final) = 40.0 L if n = 2.00 mol and T = 300 K. (c) Show the difference W(non-ideal) – W(ideal). If all your calculations are done correctly, this result shows you the effect of attractive interaction between gas particles on the work done by the system.
One mole of an ideal gas initially kept in a cylinder at pressure 1 MPa and temperature 27°C is made to expand until its volume is doubled. (a) How much work is done if the expansion is (i) adiabatic (ii) isobaric (iii) isothermal? (b) Identify the processes in which change in internal energy is least and is maximum. (c) Show each process on a PV diagram. (d) Name the processes in which the heat (Take y = 5/3 and R=8.3 J mol-1 K-1)
Knowing that the molar heat capacity at constant pressure of CO₂ has the form Cp,m = (a + b + c/T²) J mol-¹K-¹, where a= 44.22, b= 8.79 x 10-³ K-¹, and c= -8.62 x 105 K 2,calculate how much heat can absorb 1.5 moles of this gas when the temperature is increased from 25°C to 150°C. NOTE: Give your answer in kJ mol-1
(b) An ideal gas of 2.5 moles undergoes an adiabatic expansion at an initial pressure of 4.25 bar and at an initial temperature of 540 K to a final temperature of 400 K. Calculate the work done which is equal to internal energy change under such conditions. Assume Cp,m = 5/2 R %3D
2. A7.0 x 102 mol sample of Kr (g) expands reversibly and isothermally at 373 K from 5.25 mL to 6.29 mL and the internal energy of the sample is known to increase by 83.5 J. Use the virial equation of state up to the second coefficient (where Bx --28.7 mL/mol) to calculate w, q, and AH for this change. =
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