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physics
thermodynamics

Physics for Scientists and Engineers A Strategic Approach with Modern Physics 4th edition Randall D. Knight - Solutions

What fraction of monatomic molecules of a gas in a thermal equilibrium possesses kinetic energies differing from the mean value by ∂η = 1.0 % and less?
What fraction of molecules in a gas at a temperature T has the kinetic energy of translational motion exceeding έo if έo >> >>KT?
The velocity distribution of molecules in a beam coming out of a hole in a vessel is described by the function F (v) = Av3e–mv2/2hT, where T is the temperature of the gas in the vessel. Find the most probable values of (a) The velocity of the molecules in the beam; compare the result obtained
An ideal gas consisting of molecules of mass rn with concentration n has a temperature T. Using the Maxwell distribution function, find the number of molecules reaching a unit area of a wall at the angles between θ and θ + dθ} to its normal per unit time
From the conditions of the foregoing problem find the number of molecules reaching a unit area of a wall with the velocities in the interval from v to v + dv per unit time.
Find the force exerted o a particle by a uniform field if the concentrations of these particles at two levels separated by the distance Ah = 3.0 cm (along the field) differ by l = 2.0 times. The temperature of the system is equal to T = 280 K.
When examining the suspended gamboge droplets under a microscope, their average numbers in the layers separated by the distance h = 40 μm were found to differ by η = 2.0 times. The environmental temperature is equal to T = 290 K. The diameter of the droplets is d = 0.40 μm, and their
Suppose that % is the ratio of the molecular concentration of hydrogen to that of nitrogen at the Earth's surface, while η is the corresponding ratio at the height h = 3000m. Find the ratio η/ηo at the temperature T = 280 K, assuming that the temperature and the free fall
A tall vertical vessel contains a gas composed of two kinds of molecules of masses m1 and m2, with m2 > m1. The concentrations of these molecules at the bottom of the vessel are equal to n1 and n2 respectively, with n2 > n1. Assuming the temperature T and the free-fall acceleration g to be
A very tall vertical cylinder contains carbon dioxide at a certain temperature T. Assuming the gravitational field to be uniform, find how the gas pressure on the bottom of the vessel will change when the gas temperature increases η times.
A very tall vertical cylinder contains a gas at a temperature T. Assuming the gravitational field to be uniform, find the mean value of the potential energy of the gas molecules. Does this value depend on whether the gas consists of one kind of molecules or of several kinds?
A horizontal tube of length l = t00 cm closed from both ends is displaced lengthwise with a constant acceleration w. The tube contains argon at a temperature T = 330 K. At what value of w will the argon concentrations at the tube's ends differ by η = 1.0%?
Find the mass of a mole of colloid particles if during their centrifuging with an angular velocity w about a vertical axis the concentration of the particles at the distance r1 from the rotation axis is η times greater than that at the distance r2 (in the same horizontal plane). The densities
A horizontal tube with closed ends is rotated with a constant angular velocity w about a vertical axis passing through one of its ends. The tube contains carbon dioxide at a temperature T = 300 K. The length of the tube is l = 100 cm. Find the value co at which the ratio of molecular concentrations
The potential energy of gas molecules in a certain central field depends on the distance r from the field's centre as U (r) = ar2, where a is a positive constant. The gas temperature is T, the concentration of molecules at the centre of the field is no. Find: (a) The number of molecules located
From the conditions of the foregoing problem find: (a) The number of molecules whose potential energy lies within the interval from U to U + -dU; (b) The most probable value of the potential energy of a molecule; compare this value with the potential energy of a molecule located at its most
In which case will the efficiency of a Carnot cycle be higher: when the hot body temperature is increased by AT, or when the cold body temperature is decreased by the same magnitude?
Hydrogen is used in a Carnot cycle as a working substance. Find the efficiency of the cycle, if as a result of an adiabatic expansion (a) The gas volume increases n = 2.0 times; (b) The pressure decreases n = 2.0 times.
A heat engine employing a Carnot cycle with an efficiency of η = 10% is used as a refrigerating machine, the thermal reservoirs being the same. Find its refrigerating efficiency έ.
An ideal gas goes through a cycle consisting of alternate isothermal and adiabatic curves (Fig. 2.2). The isothermal processes proceed at the temperatures T1, T2, and T3. Find the efficiency of such a cycle, if in each isothermal expansion the gas volume increases in the same proportion.
Find the efficiency of a cycle consisting of two isochoric and two adiabatic lines, if the volume of the ideal gas changes n = 10 times within the cycle. The working substance is nitrogen.
Find the efficiency of a cycle consisting of two isobaric and two adiabatic lines, if the pressure changes n times within the cycle. The working substance is an ideal gas whose adiabatic exponent is equal to γ.
An ideal gas whose adiabatic exponent equals γ goes through a cycle consisting of two isochoric and two isobaric lines. Find the efficiency of such a cycle, if the absolute temperature of the gas rises n times both in the isochoric heating and in the isobaric expansion.
An ideal gas goes through a cycle consisting of(a) Isochoric, adiabatic, and isothermal lines;(b) Isobaric, adiabatic, and isothermal lines, with the isothermal process proceeding at V the minimum temperature of the whole cycle. Find the efficiency of each cycle if the absolute temperature varies
The conditions are the same as in the foregoing problem with the exception that the isothermal process proceeds at the maximum temperature of the whole cycle.
An ideal gas goes through a cycle consisting of isothermal, polytropic, and adiabatic lines, with the isothermal process proceeding at the maximum temperature of the whole cycle. Find the efficiency of such a cycle if the absolute temperature varies n-fold within the cycle.
An ideal gas with the adiabatic exponent γ goes through a direct (clockwise) cycle consisting of adiabatic, isobaric, and isochoric lines. Find the efficiency of the cycle if in the adiabatic process the volume of the ideal gas. (a) Increases n-fold; (b) Decreases n-fold.
Calculate the efficiency of a cycle consisting of isothermal, isobaric, and isochoric lines, if in the isothermal process the volume of the ideal gas with the adiabatic exponent (a) Increases n-fold; (b) Decreases n-fold.
Find the efficiency of a cycle consisting of two isochoric and two isothermal lines if the volume varies τ-fold and the absolute temperature v-fold within the cycle. The working substance is an ideal gas with the adiabatic exponent?
Find the efficiency of a cycle consisting of two isobaric and two isothermal lines if the pressure varies n-fold and the absolute temperature n-fold within the cycle. The working substance is an ideal gas with the adiabatic exponent.
An ideal gas with the adiabatic exponent γ goes through a cycle (Fig. 2.3) within which the absolute temperature varies τ-fold. Find the efficiency of this cycle.
Making use of the Clausius inequality, demonstrate that all cycles having the same maximum temperature Tmax and the same minimum temperature Tmin are less efficient compared to the Carnot cycle with the same Tmax and T min.
Making use of the Carnot theorem, show that in the case of a physically uniform substance whose state is defined by the parameters T and VWhere U (T, V) is the internal energy of the substance Instruction Consider the infinitesimal Carnot cycle in the variables p, V
Find the entropy increment of one mole of carbon dioxide when its absolute temperature increases n = 2.0 times if the process of heating is (a) Isochoric; (b) Isobaric. The gas is to be regarded as ideal.
The entropy of v ---- 4.0 moles of an ideal gas increases by ΔS = 23 J/K due to the isothermal expansion. How many times should the volume v = 4.0 moles of the gas be increased?
Two moles of an ideal gas are cooled isochorically and then expanded isobarically to lower the gas temperature back to the initial value. Find the entropy increment of the gas if in this process the gas pressure changed n = 3.3 times.
Helium of mass rn= 1.7 g is expanded adiabatically n = 3.0 times and then compressed isobarically down to the initial volume. Find the entropy increment of the gas in this process.
Find the entropy increment of v = 2.0 moles of an ideal gas whose adiabatic exponent γ = 1.30 if, as a result of a certain process, the gas volume increased a = 2.0 times while the pressure dropped β = 3.0 times,
Vessels 1 and 2 contain v = 1.2 moles of gaseous helium. The ratio of the vessels' volumes V2/V1 = a = 2.0, and the ratio of the absolute temperatures of helium in them T1/T2 = β = t.5. Assuming the gas to be ideal, find the difference of gas entropies in these vessels, S2- S1.
One mole of an ideal gas with the adiabatic exponent y goes through a polytropic process as a result of which the absolute temperature of the gas increases T-fold. The polytropic constant equals n. Find the entropy increment of the gas in this process.
The expansion process of v = 2.0 moles of argon proceeds so that the gas pressure increases in direct proportion to its volume. Find the entropy increment of the gas in this process provided its volume increases a = 2.0 times.
An ideal gas with the adiabatic exponent y goes through a process p = Po- aV, where Po and a are positive constants, and V is the volume. At what volume will the gas entropy have the maximum value?
One mole of an ideal gas goes through a process in which the entropy of the gas changes with temperature T as S = aT + CV in T, where a is a positive constant, CV is the molar heat capacity of this gas at constant volume. Find the volume dependence of the gas temperature in this process if T = To
Find the entropy increment of one mole of a Van der Waals gas due to the isothermal variation of volume from V1 to Vs. The Van der Waals corrections are assumed to be known.
One mole of a Van der Waals gas which had initially the volume V1 and the temperature T1 was transferred to the state with the volume V2 and the temperature T2. Find the corresponding entropy increment of the gas, assuming its molar heat capacity Cv to be known.
At very low temperatures the heat capacity of crystals is equal to C = aT2, where a is a constant. Find the entropy of a crystal as a function of temperature in this temperature interval.
Find the entropy increment of an aluminum bar of mass m = 3.0 kg on its heating from the temperature T1 = 300 K up to T2 = 600 K if in this temperature interval the specific heat capacity of aluminum varies as c = a + bT, where a = 0.77 J/(g . K), b = 0.46 mJ/(g . K2)
In some process the temperature of a substance depends on its entropy S as T = aSn, where a and n are constants. Find the corresponding heat capacity C of the substance as a function of S. At what condition is C < 0?
Find the temperature T as a function of the entropy S of a substance for a polytropic process in which the heat capacity of the substance equals C. The entropy of the substance is known to be equal to so at the temperature To. Draw the approximate plots T(S) for C>0 and C
One mole of an ideal gas with heat capacity Cv goes through a process in which its entropy S depends on T as S = a/T, where a is a constant. The gas temperature varies from T1 to T2. Find: (a) The molar heat capacity of the gas as a function of its temperature; (b) The amount of heat transferred to
A working substance goes through a cycle within which the absolute temperature varies n-fold, and the shape of the cycle is shown in (a) Fig. 2.4a; (b) Fig. 2.4b, where T is the absolute temperature, and S the entropy. Find the efficiency of each cycle.
One of the two thermally insulated vessels interconnected by a tube with a valve contains v = 2.2 moles of an ideal gas. The other vessel is evacuated. The valve having been opened, the gas increased its volume n = 3.0 times. Find the entropy increment of the gas.
A weightless piston divides a thermally insulated cylinder into two equal parts. One part contains one mole of an ideal gas with adiabatic exponent 7, the other is evacuated. The initial gas temperature is To. The piston is released and the gas fills the volume of the cylinder. Then the piston is
An ideal gas was expanded from the initial state to the volume V without any heat exchange with the surrounding bodies. Will the final gas pressure be the same in the case of (a) a fast and in the case of (b) a very slow expansion process?
A thermally insulated vessel is partitioned into two parts so that the volume of one part is n = 2.0 times greater than that of the other. The smaller part contains v1 = 0.30 mole of nitrogen, and the greater one v2 = 0.70 mole of oxygen. The temperature of the gases is the same. A hole is
A piece of copper of mass m1 = 300 g with initial temperature t1 = 97 °C is placed into a calorimeter in which the water of mass m2 = 100 g is at a temperature t2 = 7°C. Find the entropy increment of the system by the moment the temperatures equalize. The heat capacity of the calorimeter itself
Two identical thermally insulated vessels interconnected by a tube with a valve contain one mole of the same ideal gas each. The gas temperature in one vessel is equal to T1 and in the other, the molar heat capacity of the gas of constant volume equals the valve having been opened, the gas comes to
N atoms of gaseous helium are enclosed in a cubic vessel of volume t.0 cm 3 at room temperature. Find: (a) The probability of atoms gathering in one half of the vessel; (b) The approximate numerical value of N ensuring the occurrence of this event within the time interval t = 1010 years (the age of
Find the statistical weight of the most probable distribution of N = l0 identical molecules over two halves of the cylinder's volume. Find also the probability of such a distribution.
A vessel contains N molecules of an ideal gas. Dividing mentally the vessel into two halves A and B, find the probability that the half A contains n molecules. Consider the cases when N = 5 and n = 0, t, 2, 3, 4, 5.
A vessel of volume Vo contains N molecules of an ideal gas. Find the probability of n molecules getting into a certain separated part of the vessel of volume V. Examine, in particular, the case V = Vo/2.
An ideal gas is under standard conditions. Find the diameter of the sphere within whose volume the relative fluctuation of the number of molecules is equal to η = 1.0.10-3. What is the average number of molecules inside such a sphere?
One mole of an ideal gas consisting of monatomic molecules is enclosed in a vessel at a temperature To = 300 K. How many times and in what way will the statistical weight of this system (gas) vary if it is heated isochorically by ΔT = 1.0 K?
Find the capillary pressure. (a) In mercury droplets of diameter d = t.5 μm; (b) Inside a soap bubble of diameter d = 3.0 mm if the surface tension of the soap water solution is a = 45 mN/m.
In the bottom of a vessel with mercury there is a round hole of diameter d = 70 μm. At what maximum thickness of the mercury layer will the liquid still not flow out through this hole?
A vessel filled with air under pressure Po contains a soap bubble of diameter d. The air pressure having been reduced isothermally n-fold, the bubble diameter increased l-fold. Find the surface tension of the soap water solution.
Find the pressure in an air bubble of diameter d = 4.0 μm, located in water at a depth h = 5.0 m. The atmospheric pressure has the standard value Po.
The diameter of a gas bubble formed at the bottom of a pond is d = 5.0 μm. When the bubble rises to the surface its diameter increases n = 1.1 times. Find how deep is the pond at that spot. The atmospheric pressure is standard, the gas expansion is assumed to be isothermal.
Find the difference in height of mercury columns in two communicating vertical capillaries whose diameters are d1 = 0.50 mm and d2 = 1.00 mm, if the contact angle θ = t38 °.
A vertical capillary with inside diameter 0.50 mm is submerged into water so that the length of its part protruding over the water surface is equal to h = 25 mm. Find the curvature radius of the meniscus.
A glass capillary of length l = 110 mm and inside diameter d = 20μm is submerged vertically into water. The upper end of the capillary is sealed. The outside pressure is standard. To what length x has the capillary to be submerged to make the water levels inside and outside the capillary
When a vertical capillary of length l with the sealed upper end was brought in contact with the surface of a liquid, the level of this liquid rose to the height h the liquid density is p, the inside diameter of the capillary is d, the contact angle is 0, the atmospheric pressure is Po. Find the
A glass rod of diameter d1 = 1.5 mm is inserted sym- metrically into a glass capillary with inside diameter d2 = 2.0 mm. Then the whole arrangement is vertically oriented and brought in contact with the surface of water. To what height will the water rise in the capillary?
Two vertical plates submerged partially in a wetting liquid form a wedge with a very small angle ∂φ. The edge of this wedge is vertical. The density of the liquid is p, its surface tension is a, the contact angle is 0. Find the height h, to which the liquid rises, as a function of the
A vertical water jet flows out of a round hole. One of the horizontal sections of the jet has the diameter d = 2.0 mm while the other section located l = 20 mm lower has the diameter which is n = 1.5 times less. Find the volume of the water flowing from the hole each second.
A water drop falls in air with a uniform velocity. Find the difference between the curvature radii of the drop's surface at the upper and lower points of the drop separated by the distance h = 2.3 mm.
A mercury drop shaped as a round tablet of radius R and thickness h is located between two horizontal glass plates. Assuming that h
Find the attraction force between two parallel glass plates, separated by a distance h = 0.10 mm, after a water drop of mass m = 70 mg was introduced between them. The wetting is assumed to be complete.
Two glass discs of radius R = 5.0 cm were wetted with water and put together so that the thickness of the water layer between them was h = 1.9 μm. Assuming the wetting to be complete, find the force that has to he applied at right angles to the plates in order to pull them apart.
Two vertical parallel glass plates are partially submerged in water. The distance between the plates is d = 0.10 mm and their width is l = μ2 cm. Assuming that the water between the plates does not reach the upper edges of the plates and that the wetting is complete, find the force of their
Find the lifetime of a soap bubble of radius R connected with the atmosphere through a capillary of length l and inside radius r. The surface tension is a, the viscosity coefficient of the gas is η.
A vertical capillary is brought in contact with the water surface. What amount of heat is liberated while the water rises along the capillary? The wetting is assumed to be complete, the surface tension equals a.
Find the free energy of the surface layer of (a) A mercury droplet of diameter d = 1.4 mm; (b) A soap bubble of diameter d = 6.0 mm if the surface tension of the soap water solution is equal to a = 45 mN/m.
Find the increment of the free energy of the surface layer when two identical mercury droplets, each of diameter d = 1.5 mm, merge isothermally.
Find the work to be performed in order to blow a soap bubble of radius R if the outside air pressure is equal to P0 and the surface tension of the soap water solution is equal to a.
A soap bubble of radius r is inflated with an ideal gas. The atmospheric pressure is Po, the surface tension of the soap water solution is a. Find the difference between the molar heat capacity of the gas during its heating inside the bubble and the molar heat capacity of the gas under constant
Considering the Carnot cycle as applied to a liquid film, show that in an isothermal process the amount of heat required for the formation of a unit area of the surface layer is equal to q = T . da/dT where da/dT is the temperature derivative of the surface tension.
The surface of a soap film was increased isothermally by Δ at a temperature T. knowing the surface tension of the soap water solution a and the temperature coefficient da/dT, find the increment. (a) Of the entropy of the films surface layer; (b) Of the internal energy of the surface layer.
Calculate what fraction of gas molecules. (a) Traverses without collisions the distances exceeding the mean free path λ; (b) Has the free path values lying within the interval from λ to 2 λ.
A, narrow molecular beam makes its way into a vessel filled with gas under low pressure. Find the mean free path of molecules if the beam intensity decreases Δl-fold over the distance Al.
Let adt be the probability of a gas molecule experiencing a collision during the time interval dt a is a constant. Find: (a) The probability of a molecule experiencing no collisions during the time interval t; (b) The mean time interval between successive collisions.
Find the mean free path and the mean time interval between successive collisions of gaseous nitrogen molecules (a) Under standard conditions;(b) at temperature t = 0 °C and pressure p = 1.0 nPa'(such a pressure can be reached by means of contemporary vacuum pumps).
How many times does the mean free path of nitrogen molecules exceed the mean distance between the molecules under standard conditions?
Find the mean free path of gas molecules under standard conditions if the Van der Waals constant of this gas is equal to b = 40 ml/mol.
An acoustic we've propagates through nitrogen under standard conditions. At what frequency will the wavelength be equal to the mean free path of the gas molecules?
Oxygen is enclosed at the temperature 0 °C in a vessel with the characteristic dimension l = 10 mm (this is the linear dimension determining the character of a physical process in question). Find: (a) The gas pressure below which the mean free path of the molecules λ > l; (b) The
For the case of nitrogen under standard conditions find: (a) The mean number of collisions experienced by each molecule per second; (b) The total number of collisions occurring between the molecules within 1 cm3 a of nitrogen per second.
How does the mean free path k and the number of collisions of each molecule per unit time λ depend on the absolute temperature of an ideal gas undergoing (a) An isochoric process; (b) An isobaric process?
As a result of some process the pressure of an ideal gas increases n-fold. How many times have the mean free path k and the number of collisions of each molecule per unit time v changed and how, if the process is (a) Isochoric; (b) Isothermal?
An ideal gas consisting of rigid diatomic molecules goes through an adiabatic process. How do the mean free path λ and the number of collisions of each molecule per second v depend in this process on? (a) The volume V; (b) The pressure p; (c) The temperature T?

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