# 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}?

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}?

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?