# Basic Thermo Question (heating a room)

## Homework Statement

(This isn't for class, but still worthy of discussion!)
If you heat a room with say a fireplace, why does the total energy not increase?

PV=NkT

## The Attempt at a Solution

This is a qualitative answer (mostly). The total energy of the molecules in the room can not be changed, but I'm not sure why this is? The total energy of an ideal gas is U = (f/2)NkT
The average energy of each molecule does increase, but again why does the total energy stay the same? The answer probably has to do with PV = constant.
Again, I'm not sure, so if people have insight that would be appreciated!

## Answers and Replies

mgb_phys
Science Advisor
Homework Helper
If you heat a room with say a fireplace, why does the total energy not increase?
Are you missing a bit of the question?
The total energy of the air in the room does increase - although this is matched by the loss in the chemical energy of the fuel.

LeonhardEuler
Gold Member
There are several assumptions you are making that are not justified. First, the total energy in the room does not need to be conserved. Energy conservation only applies to isolated systems (systems where neither energy nor matter can cross the boundary)

Second, you use the energy of an ideal gas for the total energy of the room. The room does not only consist of gas, but solids as well, and particularly importantly, the logs are in the room. Also, you can not use that equation for the energy of an ideal gas if that gas goes through a reaction. That equation will tell you about how the energy changes as a result of physical changes to the state of the gas, but can't compute the energy resulting from changing from one type of gas to another.

Hopefully, if you think about why these assumptions are flawed, you will see that the question itself is flawed.

Well, lets start by saying that you are in a relatively sealed (not too much air flow) room at 290K. You wish to heat the room by lighting a fire in the fireplace. After several minutes the room is now at a more comfortable 297K.

The air in the room HAS changed temperature by 7K. The energy of the air molecules, mostly N_2 and O_2, is governed roughly by the equation:

$$E_{air} = \frac{5}{2}N k_B T$$

So the energy of the air has increased since the temperature has gone up.

However, the TOTAL energy for the air in the room, and chemical bonds of the wood in the fireplace, AND everything else in the room is the same. This being just a restatement of conservation of energy