Heating a room

1. Nov 3, 2007

mit_hacker

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. Nov 3, 2007

mjsd

how did u get this molar mass for air?

3. Nov 3, 2007

4. Nov 3, 2007

mjsd

ok, fair enough....

now, how did you work out the specific heat for air?

5. Nov 3, 2007

mjsd

how did you get your value of $$\Delta T$$?

6. Nov 3, 2007

mit_hacker

.....

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
7. Nov 4, 2007

mjsd

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)?

8. Nov 4, 2007

mit_hacker

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?

9. Nov 4, 2007

mit_hacker

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

mjsd

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?

11. Nov 4, 2007

mit_hacker

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?