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study help
physics
the physics energy
Questions and Answers of
The Physics Energy
An experimenter seeks to use a sinusoidal potential of the form studied in Example 15.2 to make a one-dimensional trap to confine a number of electrons for further study. She places one electron in
Compute the exact amplitude for transmission and reflection of a particle from a rectangular potential barrier of height V0?and width d by matching the wave function and its derivative at the
A particle of mass m and energy E encounters a triangular barrier of width 2w and height V0 (see Figure 15.6)Estimate the probability that it will tunnel through the barrier. How does the probability
Verify the barrier penetration factor obtained for the parabolic barrier in Box 15.2.
Consider the ground state and the first and second excited states of a simple harmonic oscillator as described in §7. Verify the statement made in the text that the probability of finding the
A radioactive source is emitting on average 40 particles per second. In any particular one second interval, what is the probability that it emits 40 particles? 30 particles? None? What is the
Potassium-40 (40K) is a naturally occurring form of potassium whose β-decays are responsible for some of Earth’s geothermal energy. 40K can decay in two different ways. The probabilities per unit
Uranium-238 and uranium-235 both decay by α- particle emission. Their measured half-lives are 4.5 × 109 years and 7.0 × 108 years respectively. At the present time the ratio of 235U to 238U in
The first radioactive element discovered by Polish physicist Marie Curie was radium-226. It has a halflife of 1700 years. Suppose you obtained a microgram of pure 226Ra. How long would you have to
To get a sense of the potential practical relevance for energy purposes of one aspect of as-yet-undiscovered physics, consider proton decay. Making the most general assumptions about the nature of
Einstein?s energy?momentum relation for a particle of mass m with momentum p is? Estimate the uncertainty in the x, y, and z components of the momentum of a quark confined in a (cubic) box of side
Which of the following interactions are consistent with the conservation laws for baryon number, electric charge, and the three lepton numbers? Don?t worry about energy conservation. (A bar denotes
Show that the combination of constants g2strong/ (4??0?c), where gstrong is defined in eq. (14.5) is dimensionless. It is taken as a dimensionless measure of the strength of an interaction. Compute
In principle, and in the absence of electromagnetic or any other non-gravitational interactions, an electron and a proton should form a ?gravitational atom,? bound by the force of gravity. We know
A rocket fuel should have high energy density (per kilogram) because the weight of the propellant is an important consideration. A favorite fuel is liquid hydrogen plus liquid oxygen. What is the
Verify the assertion made in §13.5.2 that the net efficiency of a CCGT is ηnet = ηB + ηR(1 − ηB), where ηB/R is the efficiency of the Brayton/Rankine cycle.
Calculate and plot the ratio of the efficiency of the Brayton cycle to the efficiency of the Otto cycle with the same compression ratio (with γ = 1.4).
Show that the ratio of the back work (the work necessary to run the compressor) to the total work done by the turbine for the ideal Brayton cycle is W{12}/W{34} = T1/T4. Check your result by
Explain why the Brayton and Otto cycles look so different in the pV-plane even though they are quite similar in the S T-plane.
The wear on a steam turbine could be decreased by raising the pressure at the turbine outlet so that the quality of the steam at the outlet is one. In the ideal Rankine cycle of Example 13.2 and
Modify the coal plant Rankine cycle described in §13.3.5 by including a regeneration cycle similar to the one shown in Figure 13.16(b)A fraction f of the steam is removed from the turbine and
Quantitatively compare the Carnot Rankine cycle of Figure 13.12(a) with the ideal Rankine cycle of Figure 13.12(b) Compare efficiencies and the work done per cycle. All the necessary information
Suppose the condenser in the 500 MWe coal power plant we analyzed in is cooled by a once through system, where water taken from the ocean is circulated once through the plant and then returned to the
When we designed the Rankine steam cycle in §13.3.5 we ignored the circulating water’s kinetic and potential energy, having claimed earlier that they are negligible. Using the work per unit mass
Sketch the ideal Rankine cycle on a pV-diagram including the saturation dome. Label the points ➀–➃. As in Figure 12.7plot the logarithm of the specific volume on the horizontal axis. 700 K 40
Neglecting the work done by the pump (which is a good approximation), show that the efficiency of the ideal Rankine cycle is η = (h3 −h4)/(h3 −h2), where hj is the specific enthalpy at
The 5-ton AC unit designed in §13.2.5 reached 33% of the Carnot limit on CoP. Look back at the definition of the “Energy Efficiency Ratio” in Problem 10.12, which employs a different temperature
A heat pump based on the ideal VC cycle of Figure 13.5(b) uses the refrigerant R-134a to heat a house. The set points are T? = ?8 ?C, T+ = 40 ?C. What is the CoP; how does it compare with the Carnot
Sketch the Carnot-like VC cycle of Figure 13.5(a) in the pV plane (superimposed on the saturation dome), and contrast the resulting shape with that of the Carnot cooling cycle. 380 360 K 340 Q(23)
Prove that the Carnot-like vapor-compression cooling cycle {1234} in Figure 13.5(a) has a maximum CoP that equals the Carnot limit regardless of the properties of the working fluid. Likewise, prove
Consider the modified Rankine cycle of Figure 13.17(a) Describe what happens to a quantity of water as it executes the cycle starting at . How many pumps and how many turbines are needed? 800 700
On a hot summer day in Houston, Texas, the day time temperature is 35 ◦C with relative humidity φ =50%. An air conditioning system cools the indoor air to 25 ◦C. How much water (kg/m3) must be
Repeat the preceding question for a sample at 10 atm with the same specific enthalpy.Preceding questionA sample of H O at a pressure of 1 atm has a specific enthalpy of 700 kJ/kg. What is its
A sample of H2O at a pressure of 1 atm has a specific enthalpy of 700 kJ/kg. What is its temperature? What state is it in, a sub-cooled liquid, super heated vapor, or a mixed phase? If it is in a
A steam turbine can be modeled as an is entropic expansion. The incoming steam is super heated at an initial temperature T+ and pressure p+. After expansion, the exhausted steam is at temperature
Revisit Question 12.2 quantitatively. Specifically, what is the pressure in the cylinder when the system comes to equilibrium after the volume has been doubled at−20 ◦C? What is the pressure
Using the data given in Table 12.1, find the deviations from the ideal gas law for water vapor just above the boiling point at p = 105 Pa. For example, does V/n = RT/p?Table 12.1 p(Tsat) 1x 10° Pa
An industrial freezer is designed to use ammonia (NH3)as a working fluid. The freezer is designed so that ammonia flowing through tubes inside the freezer at p− vaporizes at T = −40 ◦C, drawing
Data on the heat flux for laminar flow of liquid water and for pool boiling of water are shown in Figure 12.4.Take the bulk temperature of the fluid in the case of laminar flow to be T0 = 25 ◦C and
Consider a volume of 100 L of water, initially in liquid form at temperature Ti, to which ΔU = 25 MJ of energy is added from an external reservoir at temperature Tr through a thermal resistance, so
Using the data given in Table 12.1, estimate the enthalpy added to one kilogram of water that undergoes the following transformations: (a) From 0 ◦C to 90 ◦C at p = 105 Pa; (b) From 95
Check that the svap and hvap quoted in Table 12.1 are consistent with Δh = Δu + pΔv and Δs = Δh/T.Table 12.1 p(Tsat) 1x 10° Pa (99.61°C) 1.2 x 10° Pa (104.78 °C) h Temp [°C] [m*kg]
Suppose two engines, one SI, the other CI, have the same temperature range, T1 = 300K and T3 = 1800K. Suppose the SI engine, modeled as an ideal Otto cycle, has a compression ratio of 10:1, while the
A marine diesel engine has a compression ratio r = 18 and a cutoff ratio rc = 2.5. The intake air is at p1 =100 kPa and T1 = 300K. Assuming an ideal cold air standard Diesel cycle, what is the
Consider the Otto and Diesel cycles shown in Figure 11.13? The parameters have been chosen so both cycles have the same heat input and the same low temperature and pressure set point. Explain why the
Consider a throttled Otto cycle for an engine with the same parameters as Problem 11.1. In the throttled cycle assume that the spent fuel–air mixture is ejected at 1 atm and brought in again at 0.5
(Requires results from Problem 11.3) Consider the same engine as in Problem 11.3 but now run as an Atkinson cycle. The volume after expansion (V4) is still 2 L, but the volume before compression
An SI engine, modeled as an ideal Otto cycle, runs at a compression ratio of 9.6:1 with a maximum cylinder volume (V1) of 2 L and a corresponding displacement of 1.8 L. Combustion leads to a maximum
The cold air standard value of γ = 1.4 was based on the heat capacity of a diatomic gas with no vibrational excitation (see §9), CV = 5/2 nR and Cp = CV + nR. In reality, the heat capacity
Assume that an SI engine has the following parameters : displacement (V1 − V2): 2.4 L; compression ratio 9.5:1; air to fuel mass ratio 15:1; heating value of fuel 44MJ/kg, pressure at start of
Air conditioners are rated by their Energy Efficiency Ratio, or EER, defined as the cooling power Pr (in Btu/h), divided by the total electric power input Pe (in watts), measured with T+ = 95 ◦F
An air conditioner run on electricity and based on a (ideal) Carnot cycle operates between temperatures T− = 25 ◦C and T+ = 40 ◦C. The gas in the air conditioner has γ = 1.75. The AC is
Consider the cycle proposed in Question 10.3 quantitatively. Assume that it is executed reversibly. Show that its efficiency ? relative to the efficiency ?C of a Carnot engine operating between the
Mirrors are used to concentrate sunlight and heat a molten salt mixture to 500K. A heat engine is then used to convert the thermal energy to useful mechanical form. Compare a Carnot engine to a
Stirling engines generate more work per cycle than Carnot engines operating under the same conditions. Find the ratio WCarnot/W Stirling for engines run with the same compression ratio, r = V1/V3,
Consider a Carnot engine cycle operating between a maximum temperature of T3 = T4 = 400 ◦F and a minimum temperature of T1 = T2 = 60 ◦F. Assume that the lowest pressure attained is p1 = 1 atm,
Show that the changes in entropy found for iso thermal and isometric heating of an ideal gas (eqs.Andagree with the prediction from the Sackur? Tetrode equation? AS = 10 V Vi = NkBln-
Describes a thermo electric generator and mentions that the materials used should have large See beck coefficient S , small poor thermal conductivity k, and good electrical conductivity σ. Use
In isobaric heating (and expansion), heat is added reversibly to a gas kept at constant pressure. Find the change in volume and entropy when a sample of an ideal, monatomic gas, initially at
Derive eq. V2 pdV = 1 (Pi V1 – P2V2) W = AU %3D %3D 1 - NkB (T1 – T2). Y - 1
By how much does the temperature drop in the example of adiabatic expansion in Example 10.2?
Show that the method of refining tin explained in Example 9.3 will not work for alumina.
Show that roasting lead (2PbS + 3O2? 2 PbO +2 SO2) is an exothermic reaction and compute the free energy of this reaction under standard conditions. See Table for data. Compound ΔΗ AGƒ (kJ/mol)
An inexpensive hand warmer uses an exothermic chemical reaction to produce heat: iron reacts with oxygen to form ferric oxide, Fe2O3. Write a balanced chemical reaction for this oxidation process.
Thermite is a mixture of powdered metallic aluminum and iron oxide (usually Fe2O3). Although stable at room temperature, the reaction 2Al + Fe2O3 →Al2O3+ 2 Fe proceeds quickly when thermite is
Acetylene (used in welding torches) C2H2, sucrose(cane sugar) C12H22O11, and caffeine C8H10O2N4, are all popular energy sources. Their heats of combustion are 310.6 kcal/mol, 1348.2 kcal/mol, and
Estimate the lower heating value of a typical cord of wood by first estimating the mass of the wood and then assuming that this mass is completely composed of cellulose (Problem 9.16). Compare your
Wood contains organic polymers such as long chains of cellulose (C6H10O5)n, and is commonly used as abiofuel. Write a balanced equation for the complete combustion of one unit of the cellulose
Find the enthalpy of reaction for the two pathways of decomposition of TNT.
The convention we have used for LHV differs from the one offered by the US Department of Energy, which defines LHV as “the amount of heat released by combusting a specified quantity (initially at
Look up the standard enthalpies of formation and the entropies of solid ice and liquid water and verify that ice may spontaneously melt at NTP (20 ◦C, 1 atm).
Estimate the amount of CO2 produced per kilogram of CaCO3 in the calcination reaction (9.10). In addition to the CO2 released in the reaction, include the CO2 emitted if coal is burned (at 100%
Suppose reaction data, ΔH◦r and ΔG◦r (and therefore ΔS◦r), on a chemical reaction are all known at a temperature T0 and pressure p, but you want to know there action enthalpy and free energy
Show that the data in Table 9.6 satisfy the definitions of Gibbs and Helmholtz free energy, (1) ?Gr =?Hr ? T?Sr and (2) ?Hr = ?Ur + p?Vr . Quantity Value AUr Reaction internal energy +175.8 kJ/mol
Approximating the inter atomic potential as a harmonic oscillator with angular frequency ωvib, the vibrational - rotational energy levels of a diatomic molecule are given by E(n, J) =
Another commonly used model for the potential energy of a diatomic molecule is the Lennard–Jones(LJ) potential V(r) = ε((rm/r)12 − 2(rm/r)6). Using the results of Problem 9.6, find the choice of
The Morse potential parameters for oxygen are given by De = 5.211 eV, α = 2.78 Å−1, and re =1.207 Å. Using the result from the previous problem, estimate the energy necessary to excite the
Find the frequency of the lowest vibrational mode of a diatomic molecule in terms of the parameters of the Morse potential, eq. Vstone () = De (1-e -r)* -1 -a(r-re
The power output of air conditioners is measured in“tons,” an ancient nomenclature dating back to the days when air was cooled by blowing it over blocks of ice.A ton of air conditioning is
The cubic growth of the heat capacity of a solid as a function of temperature continues only up to roughly one tenth of the Debye temperature associated with the shortest wavelength excitations in
When the spin of the electron in a hydrogen atom flips from parallel to anti parallel to the direction of the protons spin, a photon of energy E ≅ 9.41 × 10−25 J is emitted. What is the
A blue super giant star may have a surface temperature as high as 50000 K. Estimate the wavelength at which it emits maximum power. Why does the star appear blue?
The universe is filled with electromagnetic radiation left over from the Big Bang. This radiation has a characteristic temperature of ≈ 2.7K. At what wavelength does the power spectrum of this
The atoms of an ideal monatomic gas of N particles confined in a box of volume V can be adsorbed onto the surface of the box (surface area A), where they are bound to the surface with binding energy
Show that the free energy of an ideal monatomic gas of N3 particles in a volume V can be written as F =−kBN3T (1 − ln(N3λ3/V)), where is a thermal length scale. Show that a two-dimensional
Show that when a system comes into thermal equilibrium while in contact with a heat reservoir at temperature T, its free energy (Helmholtz, if V is held fixed; Gibbs if p is held fixed) is minimized.
Real crystals contain impurities, which lead to nonzero entropy at T = 0. Consider a crystal consisting of N atoms. The crystal is primarily composed of element A but contains M ≪ N atoms of an
Compute the partition function per particle? for helium gas at NTP confined in a cube of side 1 cm. What quantum state has the highest probability of being occupied? What is the probability that a
It was asserted that in order for entropy to be an extensive quantity, the classical partition function of a system of indistinguishable particles must be divided by N!. Consider a box evenly
Use the Sackur?Tetrode equation?? to show that (?S/?U)|V = 1/T for an ideal monatomic gas. Likewise show that (?S/?H)|p = 1/T, verifying eq and the assertion of footnote 6. Sideal = NkB + In V + 5
Show that in general the condition that? and the requirement that the entropy be finite at all temperatures (we cannot be infinitely ignorant about a finite system), require that C(T) ? 0 as T ? 0
Check the results quoted in eqs.? for a single harmonic oscillator. Show that as T ? 0, the entropy and the heat capacity both vanish. Show that? 1 U = hw 1 ex – 1 S = kB - In(1 – e) + ex – 1
Consider a simple quantum system consisting of twenty independent simple harmonic oscillators each with frequency ω. The energy of this system is just the sum of the energies of the 20 oscillators.
Repeat the calculations of Example 8.3 for the H2?molecule, with ??/kB ? 6000K. In particular, compute the probabilities that a given molecule is in the ground state or first excited state at
A simple system in which to study entropy and the Boltzmann distribution consists of N independent two-state subsystems coupled thermally to a reservoir at temperature T. The two states in each
In a typical step in the CPU of a computer, a 32-bit register is overwritten with the result of an operation. The original values of the 32 bits are lost. Since the laws of physics are reversible,
A nuclear power plant operates at a maximum temperature of 375 ◦C and produces 1GW of electric power. If the waste heat is dumped into river water at a temperature of 20◦C and environmental
A heat engine operates between a temperature T+ and the ambient environment at temperature 298K. How large must T+ be for the engine to have a possible efficiency of 90%? Can you find some materials
Consider air to be a mixture of 78% nitrogen, 21%oxygen, and 1% argon. Estimate the minimum amount of energy that it takes to separate a cubic meter of air into its constituents at STP, by computing
Prove analytically that the entropy of a two-statesystem? is maximized when p+ = p? = 1/2. S = -kB (p+ In p+ +p_ In p_) = -kB (P+ In p4 + (1 – P+) In(1 – P+))
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