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Heating a room

  1. Nov 3, 2007 #1
    1. The problem statement, all variables and given/known data

    Imagine you've been walking outside on an cold winter's day. When you arrive home at your studio apartment, you realize that you left a window open and your 5 {\rm m} \times 5 {\rm m} \times 3 {\rm m} room is only slightly warmer than the outside. You turn on your 1-kW space heater right away and wait impatiently for the room to warm up.

    In this problem, make the following assumptions:

    * The entire 1\;{\rm kW} = 1000\;{\rm J/s} output of the space heater goes into warming the air in the room.
    * The air in the room is an ideal gas with five degrees of freedom per particle (three translational degrees of freedom and two rotational degrees of freedom--about right for nitrogen and oxygen).
    * At room temperature and atmospheric pressure, one mole of air fills a volume of 23 liters. This is slightly larger than the volume of air at standard temperature and pressure, because room temperature is hotter than 0^\circ {\rm C}.

    How long will it be before the heater warms the air in the room by 10 ^\circ {\rm C}?
    Express your answer in minutes, to the nearest integer.


    2. Relevant equations

    Q=mc(T2-T1)


    3. The attempt at a solution

    the mass of air = 29g/mol = 0.029Kg/mol.
    the volume is given to be 23 litres or 0.023m^3.
    Therefore, density = 1.26
    specific heat of air S = 1.0035 J g−1 K−1

    mass of the air enclosed in the room = desity * volume
    = 1.26 kg/m3 * 5m*5m*3m
    m = 94.5 kg
    heat reqiured to change the temperature of the room is
    Q = mSΔT
    = 94.5kg * 1003.5J kg−1 K−1 * 10 0
    = 948307.5J
    the power of the heater = 1kW
    time reqiired = t = 903150J / 1000 J /sec
    t = 948.3075 sec
    In minuted to nearest integer, this is 16.
    However, this is wrong.
    What should the correct answer be?
     
    Last edited: Nov 3, 2007
  2. jcsd
  3. Nov 3, 2007 #2

    mjsd

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    how did u get this molar mass for air?
     
  4. Nov 3, 2007 #3
  5. Nov 3, 2007 #4

    mjsd

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    ok, fair enough....

    now, how did you work out the specific heat for air?
     
  6. Nov 3, 2007 #5

    mjsd

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    how did you get your value of [tex]\Delta T[/tex]?
     
  7. Nov 3, 2007 #6
    .....

    It says in the question:

    How long will it be before the heater warms the air in the room by 10 ^\circ {\rm C}?
    Express your answer in minutes, to the nearest integer.

    Specific heat i got from another website. dont remember!!
     
    Last edited: Nov 3, 2007
  8. Nov 4, 2007 #7

    mjsd

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    while I do not know whether your specific heat value is correct or not, given that the question has told you that it is an idea gas and all that other info, I have got the feeling that you may have to work out c_v using formulas like these

    http://farside.ph.utexas.edu/teaching/sm1/lectures/node52.html

    note: it your case, there are 5 degrees of freedom for N_2 and O_2 are diatomic molecule

    anyway, a quick calculation, I've got after rounding: 11 mins.
    what is the given answer (if there is one)?
     
  9. Nov 4, 2007 #8
    Is this right:

    Cp - Cv = R

    Dividing by Cv and using the fact that Cp/Cv = gamma,

    You get Cv = R/(gamma - 1).

    Wait, whats the value of gamma for diatomic gases?
     
  10. Nov 4, 2007 #9
    Is this right:

    Cp - Cv = R

    Dividing by Cv and using the fact that Cp/Cv = gamma,

    You get Cv = R/(gamma - 1).

    Wait, whats the value of gamma for diatomic gases?
     
  11. Nov 4, 2007 #10

    mjsd

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    note once you have worked out the value of c_v it should be in units of J/mol/K
    unless your R is given in other units.

    btw, is 11 mins the correct answer?
     
  12. Nov 4, 2007 #11
    Can you please tell me the method?

    The answer is not given! Btw,

    how did u get 11?

    Cv comes to 20.755 ryt?

    then what?
     
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