AIIMS 2006 Physics Avogadro's Law MCQ Question

To analyze the question regarding the number of molecules per unit volume in two balloons filled with helium (He) gas and air, we can apply the principles of the Kinetic Theory of Gases and Avogadro's Law.

Explanation of the Correct Answer (B: same in both balloons)

According to Avogadro's Law, at the same temperature and pressure, equal volumes of gases contain an equal number of molecules, regardless of the type of gas. This law can be expressed mathematically as:

$$n \propto V$$

where $n$ is the number of moles of gas and $V$ is the volume of the gas.

In our scenario, both balloons are at the same pressure ($P$) and temperature ($T$). Since the volumes of the balloons are equal (let's denote this common volume as $V$), we can use the Ideal Gas Law, which states:

$$PV = nRT$$

Here, $R$ is the universal gas constant, and $n$ is the number of moles of gas. Rearranging the Ideal Gas Law gives:

$$n = \frac{PV}{RT}$$

From this equation, we can see that for a fixed volume $V$ and constant $R$, the number of moles $n$ is directly proportional to the pressure $P$ and inversely proportional to the temperature $T$. Since both balloons are at the same pressure and temperature, the number of moles of gas in both balloons will be the same.

To find the number of molecules per unit volume ($\rho$), we can relate it to the number of moles as follows:

$$\rho = \frac{n \cdot N_A}{V}$$

where $N_A$ is Avogadro's number ($6.022 \times 10^{23}$ molecules/mol). Therefore, since $n$ is the same for both balloons, the number of molecules per unit volume will also be the same in both balloons.

Clarification of Other Options

A) More in the He filled balloon:
This option is incorrect because, although helium is a lighter gas than air, the number of molecules per unit volume depends solely on the temperature and pressure conditions, not on the identity of the gas. Since both balloons have the same pressure and temperature, the number density of molecules will be the same.

C) More in air filled balloon:
This option is also incorrect for the same reasoning as option A. The number density of molecules is determined by the same conditions of pressure and temperature in both cases.

D) In the ratio of 1 : 4:
This option suggests a specific ratio that is not applicable here. The statement is misleading because it implies that there is a fixed numerical relationship between the two gases under the given conditions. As established, the number of molecules per unit volume is the same for both balloons due to Avogadro's Law.

Conclusion

Thus, the correct answer to the question is indeed B) same in both balloons. This conclusion follows logically from the application of Avogadro's Law and the Ideal Gas Law, confirming that the number of molecules per unit volume is independent of the type of gas when pressure and temperature are held constant.