A First Course in Differential Equations with Modeling Applications 11th edition Dennis G. Zill - Solutions

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An ideal gas is contained in a vessel at 300 K. If the temperature is increased to 900 K, by what factor does each one of the following change? (a) The average kinetic energy of the molecules. (b) The rms molecular speed. (c) The average momentum change of one molecule in a collision with a wall.
A vessel is filled with gas at some equilibrium pressure and temperature. Can all gas molecules in the vessel have the same speed?
In our model of the kinetic theory of gases, molecules were viewed as hard spheres colliding elastically with the walls of the container. Is this model realistic?
In view of the fact that hot air rises, why does it generally become cooler as you climb a mountain? (Note that air has low thermal conductivity.)
A vessel of volume V = 30 1 contains ideal gas at the temperature 0 °C. After a portion of the gas has been let out, the pressure in the vessel decreased by Δp = 0.78 atm (the temperature remaining constant). Find the mass of the released gas. The gas density under the normal conditions
Two identical vessels are connected by a tube with a valve letting the gas pass from one vessel into the other if the pressure difference Δp ≥ 1.10 atm. Initially there was a vacuum in one vessel while the other contained ideal gas at a temperature t1=27 °C and pressure ρ1 = 1.00
A vessel of volume V = 20 1 contains a mixture of hydrogen and helium at a temperature t = 20 °C and pressure p = 2.0 atm. The mass of the mixture is equal to m = 5.0 g. Find the ratio of the mass of hydrogen to that of helium in the given mixture.
A vessel contains a mixture of nitrogen (m1 = 7.0 g) and carbon dioxide (m2 = 11 g) at a temperature T = 290 K and pres- sure P0 = 1.0 atm. Find the density of this mixture, assuming the gases to be ideal.
A vessel of volume V = 7.5 1 contains a mixture of ideal gases at a temperature T = 300 K: v1 = 0.10 mole of oxygen, v2 = 0.20 mole of nitrogen, and v3 = 0.30 mole of carbon dioxide. Assuming the gases to be ideal, find: (a) The pressure of the mixture;(b) The mean molar mass M of the given mixture
A vertical cylinder closed from both ends is equipped with an easily moving piston dividing the volume into two parts, each containing one mole of air. In equilibrium at To = 300 K the volume of the upper part is η = 4.0 times greater than that of the lower part. At what temperature will the
A vessel of volume V is evacuated by means of a piston air pump. One piston stroke captures the volume ΔV. How many strokes are needed to reduce the pressure in the vessel η times? The process is assumed to be isothermal, and the gas ideal.
Find the pressure of air in a vassal being evacuated as a function of evacuation time t. The vessel volume is V, the initial pressure is/9o. The process is assumed to be isothermal, and the evacuation rata equal to C and independent of pressure. Note. The evacuation rata is the gas volume being
A chamber of volume V = 87 1 is evacuated by a pump whose evacuation rate (see Note to the foregoing problem) equals C = l0 1/s. How soon will the pressure in the chamber decrease by η = 1000 times?
A smooth vertical tube having two different sections is open from both ends and equipped with two pistons of different areas (Fig. 2.i). Each piston slides within a respective tube section. One mole of ideal gas is enclosed between the pistons tied with a non-stretchable thread. The cross-sectional
Find the maximum attainable temperature of ideal gas in each of the following processes: (a) p = po – αV2; (b) p = poe–βV, Where po, α and β are positive constants, and V is the volume of one mole of gas.
Find the minimum attainable pressure of ideal gas in the process T = To + αV2, where To and α are positive constants, and V is the volume of one mole of gas. Draw the approximate p vs V plot of this process.
A tall cylindrical vessel with gaseous nitrogen is located in a uniform gravitational field in which the free-fall acceleration is equal to g. The temperature of the nitrogen varies along the height h so that its density is the same throughout the volume. Find the temperature gradient dT/dh.
Suppose the pressure p and the density p of air are related as p/pn = const regardless of height (n is a constant hare). Find the corresponding temperature gradient.
Let us assume that air is under standard conditions close to the Earth's surface. Presuming that the temperature and the molar mass of air are independent of height, find the air pressure at the height 5.0 km over the surface and in a mine at the depth 5.0 km below the surface.
Assuming the temperature and the molar mass of air, as well as the free-fall acceleration, to be independent of the height, find the difference in heights at which the air densities at the temperature 0 °C differ (a) e times; (b) by η = 1.0%.
An ideal gas of molar mass M is contained in a tall vertical cylindrical vessel whose base area is S and height h. The temperature of the gas is T, its pressure on the bottom base is Po- Assuming the temperature and the free-fall acceleration g to be independent of the height, find the mass of gas
An ideal gas of molar mass M is contained in a very tall vertical cylindrical vessel in the uniform gravitational field in which the free-fall acceleration equals g assuming the gas temperature to be the same and equal to T, find the height at which the centre of gravity of the gas is located.
An ideal gas of molar mass M is located in the uniform gravitational field in which the free-fall acceleration is equal to g. Find the gas pressure as a function of height h, if p = P0 at h = 0, and the temperature varies with height as (a) T = To (t – ah); (b) T = To (l + oh), Where a is a
A horizontal cylinder dosed from one end is rotated with a constant angular velocity co about a vertical axis passing through the open end of the cylinder. The outside air pressure is equal to Po, the temperature to T, and the molar mass of air to M. Find the air pressure as a function of the
Under what pressure will carbon dioxide have the density ρ = 500 g/1 at the temperature T = 300 K? Carry out the calculations both for an ideal and for a Van der Waals gas.
One mole of nitrogen is contained in a vessel of volume V = 1.00 1. Find: (a) The temperature of the nitrogen at which the pressure can be calculated from an ideal gas law with an error η = 10% (as compared with the pressure calculated from the Van der Waals equation of state); (b) The gas
One mole of a certain gas is contained in a vessel of volume V = 0.250 1. At a temperature T1 = 300 K the gas pressure is Pl = 90 atm, and at a temperature T2 = 350 K the pressure is p2 = 110 atm. Find the Van der Waals parameters for this gas.
Find the isothermal compressibility × of a Van der Waals gas as a function of volume V at temperature T. Note. By definition, × = – 1/V ∂V/∂p.
Making use of the result obtained in the foregoing problem find at what temperature the isothermal compressibility u of a Van der Waals gas is greater than that of an ideal gas. Examine the case when the molar volume is much greater than the parameter b.
Demonstrate that the interval energy U of the air in a room is independent of temperature provided the outside pressure p is constant. Calculate U, if p is equal to the normal atmospheric pressure and the room's volume is equal to V = 40 ma,
A thermally insulated vessel containing a gas whose molar mass is equal to M and the ratio of specific heats Cp/Cv = γ moves with a velocity v. Find the gas temperature increment resulting from the sudden stoppage of the vessel.
Two thermally insulated vessels I and 2 are filled with air and connected by a short tube equipped with a valve. The volumes of the vessels, the pressures and. temperatures of air in them are known (V1, Pl, TI and V2, p2, T2). Find the air temperature and pressure established after the opening of
Gaseous hydrogen contained initially under standard conditions in a sealed vessel of volume V = 5.0 1 was cooled by ΔT = 55 K. Find how much the internal energy of the gas will change and what amount of heat will be lost by the gas
What amount of heat is to be transferred to nitrogen in the isobaric heating process for that gas to perform the work A = 2.0 J?
As a result of the isobaric heating by AT = 72 K one mole of a certain ideal gas obtains an amount of heat Q = 1.60 kJ. Find the work performed by the gas, the increment of its internal energy, and the value of Cp/Cv.
Two moles of a certain ideal gas at a temperature To = 300 K were cooled isochorically so that the gas pressure reduced n = 2.0 times. Then, as a result of the isobaric process, the gas expanded till its temperature got back to the initial value. Find the total amount of heat absorbed by the gas in
Calculate the value of γ = Cp/Cv- for a gaseous mixture consisting of v I = 2.0 moles of oxygen and v 2 = 3.0 moles of carbon dioxide. The gases are assumed to be ideal.
Find the specific heat capacities cv and cp, for a gaseous mixture consisting of 7.0 g of nitrogen and 20 g of argon. The gases are assumed to be ideal.
One mole of a certain ideal gas is contained under a weight-less piston of a vertical cylinder at a temperature T. The space over the piston opens into the atmosphere. What work has to be performed in order to increase isothermally the gas volume under the piston n times by slowly raising the
A piston can freely move inside a horizontal cylinder closed from both ends. Initially, the piston separates the inside space of the cylinder into two equal parts each of volume V0, in which an ideal gas is contained under the same pressure P0 and at the same temperature. What work has to be
Three moles of an ideal gas being initially at a temperature To = 273 K were isothermally expanded n = 5.0 times its initial volume and then isochorically heated so that the pressure in the final state became equal to that in the initial state. The total amount of heat transferred to the gas during
Draw the approximate plots of isochoric, isobaric, isothermal, and adiabatic processes for the case of an ideal gas, using the following variables: (a) p, T; (b) V, T.
One mole of oxygen being initially at a temperature To = 290 K is adiabatically compressed to increase its pressure η = 10.0 times. Find: (a) The gas temperature after the compression; (b) The work that has been performed on the gas.
A certain mass of nitrogen was compressed η = 5.0 times (in terms of volume), first adiabatically, and then isothermally. In both cases the initial state of the gas was the same. Find the ratio of the respective works expended in each compression.
A heat-conducting piston can freely move inside a closed thermally insulated cylinder with an ideal gas. In equilibrium the piston divides the cylinder into two equal parts, the gas temperature being equal to To. The piston is slowly displaced. Find the gas temperature as a function of the ratio
Find the rate v with which helium flows out of a thermally insulated vessel into vacuum through a small hole. The flow rate of the gas inside the vessel is assumed to be negligible under these conditions. The temperature of helium in the vessel is T = 1,000 K.
The volume of one mole of ∆n ideal gas with the adiabatic exponent γ is varied according to the law V = a/T, where a is a constant. Find the amount of heat obtained by the gas in this process if the gas temperature increased by ΔT.
Demonstrate that the process in which the work performed by an ideal gas is proportional to the corresponding increment of its internal energy is described by the equation pVn = const, where n is a constant.
Find the molar heat capacity of an ideal gas in a polytropic process pVn = const if the adiabatic exponent of the gas is equal to γ. At what values of the polytropic constant n will the heat capacity of the gas be negative?
In a certain polytropic process the volume of argon was in- creased a- 4.0 times. Simultaneously, the pressure decreased β = 8.0 times. Find the molar heat capacity of argon in this process, assuming the gas to be ideal.
One mole of argon is expanded polytropically, the polytropic constant being n----1.50. In the process, the gas temperature changes by ΔT = – 26 K. Find: (a) The amount of heat obtained by the gas; (b) The work performed by the gas.
n ideal gas whose adiabatic exponent equals γ is expanded according to the law p = αV, where a is a constant. The initial volume of the gas is equal to Vo, as a result of expansion the volume in-creases B times. Find: (a) The increment of the internal energy of the gas; (b) The work
An ideal gas whose adiabatic exponent equals γ is expanded so that the amount of heat transferred to the gas is equal to the decrease of its internal energy. Find: (a) The molar heat capacity of the gas in this process; (b) The equation of the process in the variables T, V; (c) The work
One mole of an ideal gas whose adiabatic exponent equals γ undergoes a process in which the gas pressure relates to the temperature as p = aTa, where a and a are constants. Find: (a) The work performed by the gas if its temperature gets an increment ΔT; (b) The molar heat capacity of
An ideal gas with the adiabatic exponent γ undergoes a process in which its internal energy relates to the volume as U = aVα, where a and a are constants. Find: (a) The work performed by the gas and the amount of heat to be transferred to this gas to increase its internal energy by AU;
An ideal gas has a molar heat capacity Cv at constant volume. Find the molar heat capacity of this gas as a function of its volume V, if the gas undergoes the following process: (a) T = Toe αv ; (b) p = poeαv , Where To, P0, and a are constants.
One mole of an ideal gas whose adiabatic exponent equals γ undergoes a process p = P0 + a/V, where P0 and a are positive constants. Find: (a) Heat capacity of the gas as a function of its volume; (b) The internal energy increment of the gas, the work performed by it, and the amount of heat
One mole of an ideal gas with heat capacity at constant pressure Cp undergoes the process T = To + aV, where To and a are constants. Find: (a) Heat capacity of the gas as a function of its volume; (b) The amount of heat transferred to the gas, if its volume in-creased from V1 to V2.
For the case of an ideal gas find the equation of the process (in the variables T, V) in which the molar heat capacity varies as: (a) C = Cv + aT; (b) C = Cv + βV; (e) C = Cv + ap, Where a, β, and a are constants.
An ideal gas has an adiabatic exponent y. In some process its molar heat capacity varies as C = a/T, where a is a constant. Find: (a) The work performed by one mole of the gas during its heating from the temperature To the temperature η times higher; (b) The equation of the process in the
Find the work performed by one mole of a Van der Waals gas during its isothermal expansion from the volume V1 to V2 at a temperature T.
One mole of oxygen is expanded from a volume V1 = 1.00 1 to V2 = 5.0 l at a constant temperature T = 280 K. Calculate: (a) The increment of the internal energy of the gas: (b) The amount of the absorbed heat. The gas is assumed to be a Van der Waals gas.
For a Van der Waals gas find: (a) The equation of the adiabatic curve in the variables T, V; (b) The difference of the molar heat capacities Cp = Cv as a function of T and V.
Two thermally insulated vessels are interconnected by a tube equipped with a valve. One vessel of volume V1 = 10 1 contains v = 2.5 moles of carbon dioxide. The other vessel of volume V2 = 100 1 is evacuated. The valve having been opened, the gas adiabatically expanded. Assuming the gas to obey the
What amount of heat has to be transferred to v = 3.0 moles of carbon dioxide to keep its temperature constant while it expands into vacuum from the volume V1 = 5.0 1 to V2 = 10 1? The gas is assumed to be a Van der Waals gas.
Modern vacuum pumps permit the pressures down to p = 4.10-15 atm to be reached at room temperatures. Assuming that the gas exhausted is nitrogen, find the number of its molecules per 1 cm 3 and the mean distance between them at this pressure.
A vessel of volume V = 5.0 1 contains m = 1.4 g of nitrogen t a temperature T = 1800 K. Find the gas pressure, taking into account that η = 30% of molecules are disassociated into atoms at this temperature.
Under standard conditions the density of the helium and nitrogen mixture equals ρ = 0.60 g/1. Find the concentration of helium atoms in the given mixture.
A parallel beam of nitrogen molecules moving with velocity v = 400 m/s impinges on a wall at an angle θ = 30 ° to its normal. The concentration of molecules in the beam n = 0.9.1019 cm–3. Find the pressure exerted by the beam on the wall assuming the molecules to scatter in accordance with
How many degrees of freedom have the gas molecules, if under standard conditions the gas density is ρ = 1.3 mg/cm s and the velocity of sound propagation in it is v = 330 m/s.
Determine the ratio of the sonic velocity v in a gas to the root mean square velocity of molecules of this gas, if the molecules are (a) Monatomic; (b) Rigid diatomic.
A gas consisting of N-atomic molecules has the temperature T at which all degrees of freedom (translational, rotational, and vibrational) are excited. Find the mean energy of molecules in such a gas. What fraction of this energy corresponds to that of translational motion?
Suppose a gas is heated up to a temperature at which all degrees of freedom (translational, rotational, and vibrational) of its molecules are excited. Find the molar heat capacity of such a gas in the isochoric process, as well as the adiabatic exponent γ, if the gas consists of. (a)
An ideal gas consisting of N-atomic molecules is expanded isobarically. Assuming that all degrees of freedom (translational, rotational, and vibrational) of the molecules are excited, find what fraction of heat transferred to the gas in this process is spent to perform the work of expansion. How
Find the molar mass and the number of degrees of freedom of molecules in a gas if its heat capacities are known: Cg = = 0.65 J/(g . K) and cp = 0.91 J/(g . K).
Find the number of degrees of freedom of molecules in a gas whose molar heat capacity (a) At constant pressure is equal to Cp = 29 J/(mol. K); (b) Is equal to C = 29 J/(mol. K) in the process pT = const.
Find the adiabatic exponent γ for a mixture consisting of v1 moles of a monatomic gas and v2 moles of gas of rigid diatomic molecules.
A thermally insulated vessel with gaseous nitrogen at a temperature t = 27 °C moves with velocity v = 100 m/s. How much (in per cent) and in what way will the gas pressure change on a sudden stoppage of the vessel?
Calculate at the temperature t = i7 °C: (a) The root mean square velocity and the mean kinetic energy of an oxygen molecule in the process of translational motion; (b) The root mean square velocity of a water droplet of diameter d = 0.10 μm suspended in the air.
A gas consisting of rigid diatomic molecules is expanded adiabatically. How many times has the gas to be expanded to reduce the root mean square velocity of the molecules η = t.50 times?
The mass m = 15 g of nitrogen is enclosed in a vessel at a temperature T = 300 K. What amount of heat has to be transferred to the gas to increase the root mean square velocity of its molecules η = 2.0 times?
The temperature of a gas consisting of rigid diatomic molecules is T = 300 K. Calculate the angular root mean square velocity of a rotating molecule if its moment of inertia is equal to I = 2.1.10-39 g . cm%
A gas consisting of rigid diatomic molecules was initially under standard conditions. Then the gas was compressed adiabatically η = 5.0 times. Find the mean kinetic energy of a rotating molecule in the final state.
How will the rate of collisions of rigid diatomic molecules against the vessel's wall change, if the gas is expanded adiabatically η times?
The volume of gas consisting of rigid diatomic molecules was increased η = 2.0 times in a polytropic process with the molar heat capacity C = R. How many times will the rate of collisions of molecules against a vessel's wall be reduced as a result of this process?
A gas consisting of rigid diatomic molecules was expanded in a polytropic process so that the rate of collisions of the molecules against the vessel's wall did not change. Find the molar heat capacity of the gas in this process.
Calculate the most probable, the mean, and the root mean square velocities of a molecule of a gas whose density under standard atmospheric pressure is equal to ρ = 1.00 g/1.
Find the fraction of gas molecules whose velocities differ by less than ∂η = 1.00% from the value of. (a) The most probable velocity; (b) The root mean square velocity.
Determine the gas temperature at which (a) The root mean square velocity of hydrogen molecules exceeds their most probable velocity by Av = 400 m/s; (b) The velocity distribution function F (v) for the oxygen molecules will have the maximum value at the velocity v = 420 m/s.
In the case of gaseous nitrogen find: (a) The temperature at which the velocities of the molecules v1 = 300 m/s and v2 = 600 m/s are associated with equal values of the Maxwell distribution function F (v); (b) The velocity of the molecules v at which the value of the Maxwell distribution
At what temperature of a nitrogen and oxygen mixture do the most probable velocities of nitrogen and oxygen molecules differ by Δv = 30 m/s?
The temperature of a hydrogen and helium mixture is T = 300 K. At what value of the molecular velocity v will the Maxwell distribution function F (v) yield the same magnitude for both gases?
At what temperature of a gas will the number of molecules, whose velocities fall within the given interval from v to v + dv, be the greatest? The mass of each molecule is equal to m.
Find the fraction of molecules whose velocity projections on the x axis fall within the interval from vx to v1 + dy1,, while the moduli of perpendicular velocity components fall within the interval from v ± to v + q- dr1. The mass of each molecule is m, and the temperature is T.
Using the Maxwell distribution function, calculate the mean velocity projection (vx) and the mean value of the modulus of this projection (| vx |) if the mass of each molecule is equal to m and the gas temperature is T.
From the Maxwell distribution function find (v2x), the mean value of the squared vx projection of the molecular velocity in a gas at a temperature T. The mass of each molecule is equal to m.
Making use of the Maxwell distribution function, calculate the number v of gas molecules reaching a unit area of a wall per unit time, if the concentration of molecules is equal to n, the temperature to T, and the mass of each molecule is m.
Using the Maxwell distribution function, determine the pressure exerted by gas on a wall, if the gas temperature is T and the concentration of molecules is n.
Making use of the Maxwell distribution function, find (1/v), the mean value of the reciprocal of the velocity of molecules in an ideal gas at a temperature T, if the mass of each molecule is equal to m. Compare the value obtained with the reciprocal of the mean velocity.
A gas consists of molecules of mass m and is at a temperature T. Making use of the Maxwell velocity distribution function find the corresponding distribution of the molecules over the kinetic energies έ. Determine the most probable value of the kinetic energy έp. Does έp correspond

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A First Course in Differential Equations with Modeling Applications 11th edition Dennis G. Zill - Solutions

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