An Introduction to Thermal Physics
1st Edition
ISBN: 9780201380279
Author: Daniel V. Schroeder
Publisher: Addison Wesley
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Chapter 4.2, Problem 16P
To determine
Proof of the fact that a
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Equations 6.92 and 6.93 (See attached) for the entropy and chemical potential involve the logarithm of the quantity V Zint!N vQ. Is this logarithm normally positive or negative? Plug in some numbers for an ordinary gas and discuss.
Now, let's use this property of logarithms to learn something about the number of microstates available to a molecular system. The absolute
entropy of a system is related to the number of microstates available to it via Boltzmann's formula S = kB In W. If a system containing one
mole of an ideal gas has an entropy of 167.7 J/K, how many microstates does it have? Report the order of W, as we have defined it above,
and you should use scientific notation, 1.23E45, and report 3 (three) significant figures.
2.2
Consider a reversible process that takes an ideal gas
from an initial state a to a final equilibrium state b. Show that the
change in entropy of the gas is given by:
AS = nC, In ()+nR In ()
CL
Chapter 4 Solutions
An Introduction to Thermal Physics
Ch. 4.1 - Prob. 1PCh. 4.1 - At a power plant that produces 1 GW ( 109 watts)...Ch. 4.1 - A power plant produces 1 GW of electricity, at an...Ch. 4.1 - It has been proposed to use the thermal gradient...Ch. 4.1 - Prove directly (by calculating the heat taken in...Ch. 4.1 - To get more than an infinitesimal amount of work...Ch. 4.2 - Why must you put an air conditioner in the window...Ch. 4.2 - Can you cool off your kitchen by leaving the...Ch. 4.2 - Prob. 9PCh. 4.2 - Suppose that heat leaks into your kitchen...
Ch. 4.2 - What is the maximum possible COP for a cyclic...Ch. 4.2 - Explain why an ideal gas taken around a...Ch. 4.2 - Under many conditions, the rate at which heat...Ch. 4.2 - Prob. 14PCh. 4.2 - In an absorption refrigerator the energy driving...Ch. 4.2 - Prob. 16PCh. 4.2 - Prob. 17PCh. 4.3 - Prob. 18PCh. 4.3 - The amount of work done by each stroke of an...Ch. 4.3 - Derive a formula for the efficiency of the Diesel...Ch. 4.3 - The ingenious Stirling engine is a true heat...Ch. 4.3 - A small-scale steam engine might operate between...Ch. 4.3 - Prob. 23PCh. 4.3 - Calculate the efficiency of a Rankine cycle that...Ch. 4.3 - In a real turbine, the entropy of the steam will...Ch. 4.3 - A coal-fired power plant, with parameters similar...Ch. 4.3 - In Table 4.1, why does the entropy of water...Ch. 4.3 - Imagine that your dog has eaten the portion of...Ch. 4.4 - Liquid HFC-134a at its boiling point at 12 bars...Ch. 4.4 - Consider a household refrigerator that uses...Ch. 4.4 - Suppose that the throttling valve in the...Ch. 4.4 - Suppose you are told to design a household air...Ch. 4.4 - Prob. 33PCh. 4.4 - Consider an ideal Hampson-Linde cycle in which no...Ch. 4.4 - The magnetic field created by a dipole has a...Ch. 4.4 - Prob. 36PCh. 4.4 - A common (but imprecise) way of stating the third...
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- Q.17 Consider two identical, finite, isolated systems of constant heat capacity C at temperatures T, and T2 (T> T:). An engine works between them until their temperatures become equal. Taking into account that the work performed by the engine will be maximum (= Wmax) if the process is reversible (equivalently, the entropy change of the entire system is zero), the value of Wmax is: (A) C(T, – T2) (B) C (T, – T2)/2 (C) C(T, + T2 -- T;T2) (D) C(/T, - T)arrow_forwardAt steady state, a thermodynamic cycle operating between hot and cold reservoirs at 1000 K and 500 K, respectively, receives energy by heat transfer from the hot reservoir at a rate of 1500 kW, discharges energy by heat transfer to the cold reservoir, and develops power at a rate of (a) 1000 kW, (b) 750 kW, (c) 0 kW. For each case, apply Eq. 5.13 on a time- rate basis to determine whether the cycle operates reversibly, operates irreversibly, or is impossible.arrow_forward(a) Answer the following questions about entropy-volume relationships: (i) For a general system, use the Helmholtz free energy as an intermediary to express the derivative () in terms of a T. derivative of the pressure P with respect to temperature T. (ii) Assuming an ideal gas, evaluate your derivative of P, and finally integrate () to determine the volume dependence of the av entropy S of the classical ideal gas. (iii) Comment on your result, and in particular on how an alternative understanding of the S(V) dependence can be achieved on the basis of spatial multiplicity considerations.arrow_forward
- Prove that the entropy S of an ideal gas [Sackur and Tetrode's equation] is an extensive quantity. Then show that the entropy of the gas of particles to be separated from each other is 3 S = NKB — Nku [2 - In (2/V)], and that this quantity is not extensive. Remember: by extensiveness we mean that if we scale the size of the system by a factor a (V → a V, N a N, but the particle density n = N/V remains constant), any extensive quantity a s) also scales by a factor a (here: S →arrow_forwardConsider a van der Waal's gas that undergoes an isothermal expansion from volume V₁ to volume V₂. Calculate the change in the Helmholtz free energy. 2.2 (a) (b) From the theory of thermodynamics, with T and V T independent, ()₁ = T()-p. Show that the change in internal energy is AU = a(-1/2).arrow_forwardA plastic bag containing 0.2 kg of water at 20°C is dropped from a height of 0.5 m onto an insulating carpet. Assume that the bag does NOT break. What is the approximate probability that a similar bag sitting on a carpet will do the reverse; that is, spontaneously jump 0.5 m in the air? Express your answer in the form "Probability = 10-x," where x is a number you will calculate. (Hint: Note that ey = 10y÷ln(10).)arrow_forward
- 3.2 Show that the entropy maximum principle encountered in the study of isolated systems [eq. (2.50)] is equivalent to the statement that the non- flow exergy of an isolated system is destined to decrease or, at best, remain unchanged. 33 Conuidarrow_forward8.6. A current of 10 A is maintained for 1s in a resistor of 25 N while the temperature of the resistor is kept constant at 27°C. (a) What is the entropy change of the resistor? (b) What is the entropy change of the universe? The same current is maintained for the same time in the same resistor, but now thermally insulated, with the same initial temperature. If the resistor has a mass of 10g and a specific heat of 836 J/kg - K: (c) What is the entropy change of the resistor? (d) What is the entropy change of the universe?arrow_forwardConsider again the system of two large, identical Einstein solids. For the case N = 1023 , compute the entropy of this system (in terms of Boltzmann's constant), assuming that all of the microstates are allowed. (This is the system's entropy over long time scalesarrow_forward
- An ideal gas initially at P, V, and T, is taken through a cydle as shown below. (Let the factor n - 3.3.) P B P, V. (a) Find the net work done on the gas per cycle for 2.45 mol of gas initially at 0°C. kJ (b) What is the net energy added by heat to the system per cycle?arrow_forwardConsider a system of two Einstein solids, with NA = 300, NB = 200, and qtotal = 100 Compute the entropy of the most likely macrostate and of the least likely macrostate. Also compute the entropy over long time scales, assuming that all microstates are accessible. (Neglect the factor of Boltzmann's constant in the definition of entropy; for systems this small it is best to think of entropy as a pure number.)arrow_forwardIn Debye Approximation the entropy at some temperature T (less than 10 K) is aT(Blank 1 ) /3 If the value of this entropy at T = 3.5 K is 1.55 J / K , then the value of the coefficient "a" is : ( Blank 2)arrow_forward
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