Animated Solution for Chemistry - States of Matter: An open vessel at $27^{\circ }\text{C}$ is heated until two fifth of the air (assumed as an ideal gas) in it has escaped from the vessel. Assuming that the volume of the vessel remains constant, the temperature at which the vessel has been heated is

Imagine you have an open vessel sitting on a table. The temperature in the room is $27^{\circ }\text{C}$. Because the vessel is open, the gas inside is in direct contact with the atmosphere. This means the pressure inside the vessel will always be equal to the atmospheric pressure. So, pressure ($P$) is constant. The vessel itself is rigid, so its volume ($V$) is also constant.

Let's quickly convert our starting temperature to Kelvin by adding $273$, giving us an initial temperature $T_1=300\text{ K}$.

Now, let's bring in our trusty tool, the ideal gas equation:

We just established that pressure $P$ and volume $V$ are constant. And of course, the universal gas constant $R$ is always constant. If we rearrange the equation to solve for $n$, we get:

Since $P$, $V$, and $R$ are all constants, we can clearly see that the number of moles $n$ is inversely proportional to the temperature $T$. This gives us a beautiful relationship:

This is the master key to solving our problem.

The problem tells us that the vessel is heated until $\frac{2}{5}$ of the air escapes. Let's say initially, we had $n$ moles of air inside the vessel. That's our $n_1$.

If $\frac{2}{5}$ of those moles escape into the atmosphere, how many moles are left inside? We have to be careful here; our gas law applies to the gas remaining inside the vessel. So, we subtract $\frac{2}{5}n$ from the initial $n$:

This remaining amount is our final number of moles, $n_2$.

We have all our pieces; now let's put them together. We substitute our values into the master equation:

Look at the equation now. We have the unknown variable $n$ on both sides. This is a classic physics problem trick! We don't even need to know the exact initial number of moles because $n$ will simply cancel out.

After canceling $n$ from both sides, we are left with:

To find $T_2$, we need to isolate it. We multiply $300$ by $5$ and then divide by $3$:

Before we wrap up, think about the core concept here. In any open vessel problem, the pressure and volume are your constants, leading to moles being inversely proportional to temperature. If the question had asked for the answer in Celsius, we would just subtract $273$ from $500$ to get $227^{\circ }\text{C}$. Always pay attention to the units in the options!