Physics for Scientists and Engineers with Modern Physics, Technology Update
Physics for Scientists and Engineers with Modern Physics, Technology Update
9th Edition
ISBN: 9781305401969
Author: SERWAY, Raymond A.; Jewett, John W.
Publisher: Cengage Learning
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Chapter 20, Problem 81CP

(a)

To determine

The speed of rise of the piston when the water begins boiling.

(a)

Expert Solution
Check Mark

Answer to Problem 81CP

The speed of rise of the piston when the water begins boiling is 4.19 mm/s.

Explanation of Solution

A piston-cylinder apparatus of cylinder radius r=7.50 cm and piston mass of m=3.00 kg sits on the surface of water at the bottom of the cylinder of 2.00 kg at temperature just below 100.0°C. Now, an electric heater in the cylinder base is turned on and it transfers energy to the water at the rate of 100 W.

The speed of rise of the piston is the increase in the height h due to the boiling of water and the steam produced in the process. The volume of the steam or gas is the area A of the cylinder times the height h

    v = dhdt = ddt(VA)

Using ideal gas law, PV=nRT in the above equation and solving

  v = 1Addt(nRTP)

  v= RTPAdndt                                                                       (I)

Here, PA is the force applied by the gas on the piston, v is the volume of the gas, n is the number of moles, T is the temperature and R is the ideal gas constant.

Write the formula for the force when the piston is in equilibrium

  F=P0A+mpg                                                              (II)

Here, P0 is the atmospheric pressure, A is the area, mp is the mass of the piston and g is the acceleration due to gravity.

Write the formula to find the number of moles

      n=mgMw                                                              (III)

Here, mg is the mass of gas and Mw is the molecular mass.

Write the formula to find energy required to change the phase

    Q=LvΔm                                                                              (IV)

Here, Q is the energy required to change the phase, Lv is the latent heat of vaporization and Δm is the change in mass.

Substitute equation (II) and (III) in (I) and solving

  v = (RTmpg + P0A)ddt(mgMw) = [RT(mpg + P0A)Mw]dmgdt

Substitute equation (IV) in the above equation

v = RT(mpg + P0A)Mwddt(QLv)= RT(mpg + P0A)MwLvdQdt

Here, dQdt is power. Therefore,

v = RT(Power)(mpg + P0A)MwLv

Substitute 8.314 J/molK for R, 373 K for T, 100 for P, 3.00 kg for mp, 9.80 m/s2 for g, 1.013×105 Pa for P0, 7.50 cm for r, 0.0180 for Mw and 2.26×106 J/kg for Lv in the above equation to find the value of v

v = (8.314)(373)(100)[(3.00)(9.80)+(1.013×105)(π)(0.0750)2](0.0180)(2.26×106)   = 4.19 × 103 m/s =4.19 mm/s

Thus, the speed of rise of the piston when the water begins boiling is 4.19 mm/s.

(b)

To determine

The speed of rise of the piston when the water is completely turned into steam.

(b)

Expert Solution
Check Mark

Answer to Problem 81CP

The speed of rise of the piston when the water is completely turned into steam is 12.6 mm/s.

Explanation of Solution

When the water in the cylinder is completely turned into steam, the number of moles will be fixed and the temperature changes as the heater continues to transfer energy to the steam, equation (I) becomes

    v = (nRPA)dTdt v= nR(mpg + P0A)dTdt 

Substitute equation (IV) in the above equation

v = nR(mpg + P0A)ddt(Qmgc) = mgR(mpg + P0A)mgcMwdQdt = R(Power)(mpg + P0A)cMw

Substitute 8.314 J/molK for R, 100 for P, 3.00 kg for mp, 9.80 m/s2 for g, 1.013×105 Pa for P0, 7.50 cm for r, 0.0180 for Mw and 2.26×106 J/kg for Lv in the above equation to find the value of v

v = (8.314)(100)[(3.00)(9.80)+(1.013×105)(π)(0.0750)2](2 010)(0.018 0)    = 0.012 6 m/s =12.6 mm/s

Thus, the speed of rise of the piston when the water is completely turned into steam is 12.6 mm/s.

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Chapter 20 Solutions

Physics for Scientists and Engineers with Modern Physics, Technology Update

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