Animated Solution for Physics - Thermodynamics: The temperature of an open room of volume $30{\text{ m}}^3$ increases from $17^{\circ }\text{C}$ to $27^{\circ }\text{C}$ due to the sunshine. The atmospheric pressure in the room remains $1\times 10^5\text{ Pa}$. If $n_i$ and $n_f$ are the number of molecules in the room before and after heating, then $n_f-n_i$ will be

Imagine you are sitting in a room on a bright, sunny day. The volume of the room is fixed at $30{\text{ m}}^3$. Because the room has open windows or doors, it is not a sealed container. This means it is in constant equilibrium with the Earth's atmosphere, so the pressure inside remains perfectly constant at $1\times 10^5\text{ Pa}$.

However, as the sun beats down, the temperature inside the room rises from $17^{\circ }\text{C}$ to $27^{\circ }\text{C}$. What happens to the air inside? Does the number of molecules stay the same? Let's dive into the mathematics of the Kinetic Theory of Gases to find out.

Adapting the Ideal Gas Law

We start with the fundamental Ideal Gas Law:

Here, $n$ represents the number of moles of the gas. But our question asks about the actual number of molecules, denoted as $N$. We know that one mole contains Avogadro's number ($N_A$) of molecules. Therefore, the number of moles is simply the total number of molecules divided by Avogadro's number:

Substituting this back into our ideal gas equation gives us:

Rearranging this to solve for the number of molecules $N$, we get our master equation for this problem:

The problem asks us to find the change in the number of molecules, specifically $n_f-n_i$ (final molecules minus initial molecules). Using our master equation, we can write expressions for both the initial and final states:

Notice something beautiful here? The pressure $p$, the volume $V$, Avogadro's number $N_A$, and the universal gas constant $R$ are all constants. The only thing changing is the temperature. Let's factor out the constants to make our calculation elegant:

Before we plug in the numbers, we must avoid a classic trap: Temperatures must always be in Kelvin! Using Celsius in thermodynamics will lead to physically impossible results.

Initial temperature: $T_i=17+273=290\text{ K}$ Final temperature: $T_f=27+273=300\text{ K}$

Now, let's substitute all our known values into the factored equation:

Let's carefully evaluate the bracketed term by finding a common denominator:

Now, we multiply everything together:

We arrived at a final answer of $-2.5\times 10^{25}$. But what does that negative sign actually mean in the real world?

A negative change ($\Delta n<0$) means that the final number of molecules is less than the initial number. As the sun heated the air inside the room, the kinetic energy of the gas molecules increased, causing the air to expand. Because the room is open and the volume is fixed, this expanding air had nowhere to go but out the window.

Thus, $2.5\times 10^{25}$ molecules physically left the room. The mathematics flawlessly captured the physical reality of thermal expansion in an open system!