# Pressure, density, volume and mass in a room

Here's the question.

A window in the room is open. The next day, the temperature of the room has increased, but the pressure of the air stayed the same. State and explain what has happened to the mass of air in the room.

The model answer , which I don't understand really well: mass (of air) in room decreases B1 (because) air expands / vol of air increases / density of air decreases / appropriate use of pV = nRT OR pressure argument e.g. pressure would have increased (with constant volume) if mass constant B1 any ONE from: B1 some air leaves room molecules collide harder or more (often) molecules move faster / have more energy molecules move further apart NOT molecules expand

I'd appreciate it if someone could explain it because the model answer is very misleading.

Assuming we can treat the air in the room as an ideal gas, it will obey the ideal gas equation of state:

$$PV = nRT \tag{1}$$

where $n$ is the number of moles of the gas.

The question tells us that the pressure is constant, and obviously the volume of the room is constant, so the only things that can vary are $T$ and $n$. The question tells us that $T$ has changed, so let's rearrange equation (1) to give $n$ as a function of $T$:

$$n = \frac{PV}{R}\frac{1}{T} \tag{2}$$

or since $P$, $V$ and $R$ are all constants:

$$n \propto \frac{1}{T}$$

So as the temperature in the room goes up the mass of air in the room goes down in inverse proportion.

I find the statements you've attributed to the model answer too vague to usefully comment on. The answer above is the only one I would accept if I were marking the exam. If you really wanted to put it into words I suppose I'd say the increase in temperature made the air in the room expand so it would no longer all fit in the room and some had to leave through the window.