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imagine that 501 people are present in a movie theater of volume 8.00 x10^3 that is sealed shut so no air can escape. Each person gives off heat at an average rate of 110 W. By how much will the temperature of the air have increased during a 2.0 h movie? The initial pressure is 1.01*10^5 Pa and the initial temperature is 20.0 degrees celcius

Respuesta :

Answer:

The temperature of air will increase by [tex]\Delta T=41044.967\ K[/tex]

Explanation:

Given:

  • no. of person in a theater, [tex]n=501[/tex]
  • volume of air in the theater, [tex]V=8\times 10^3\ m^3[/tex]
  • rate of heat given off by each person, [tex]P=110\ J.s^{-1}[/tex]
  • duration of movie, [tex]t=2\ hr=7200\ s[/tex]
  • initial pressure in the theater, [tex]p_i=1.01\times 10^5\ Pa[/tex]
  • initial temperature in the theater, [tex]T_i=20+273=293\ K[/tex]
  • specific heat capacity of air at the given conditions, [tex]c=1.0061\ J.kg^{-1}.K^{-1}[/tex]

The total quantity of heat released by the total people in the theater during the movie:

[tex]Q=n.P.t[/tex]

[tex]Q=501\times 110\times 7200[/tex]

[tex]Q=396792000\ J[/tex]

Form the relation of heat capacity:

[tex]Q=m.c.\Delta T[/tex]

∵[tex]p_i.V=m.R.T[/tex]

[tex]Q=(\frac{p_i.V}{R.T}) \times c\times (T_f-T_i)[/tex]

[tex]396792000=(\frac{1.01\times 10^5\times 8\times 10^3}{287\times 293}) \times 1.0061\times (T_f-293)[/tex]

[tex]T_f=41337.967\ K[/tex]

Change in temperature of air:

[tex]\Delta T=41044.967\ K[/tex]