AIIMS 2006 Physics Ideal Gas Law MCQ Question

To solve the problem, we need to apply the Ideal Gas Law, which is given by the formula:

$$PV = nRT$$

where:

• $P$ is the pressure,
• $V$ is the volume,
• $n$ is the number of moles of gas,
• $R$ is the universal gas constant,
• $T$ is the temperature in Kelvin.

Explanation of the Correct Answer

Given that both balloons are at the same pressure $P$ and temperature $T$, we can derive the number of molecules per unit volume (which is the number density of the gas) in each balloon.

1. Calculating Number of Moles: From the Ideal Gas Law, we can rearrange the equation to find the number of moles $n$:

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

2. Calculating Number of Molecules: The number of molecules $N$ can be calculated from the number of moles using Avogadro's number $N_A$:

   $$N = n \times N_A = \frac{PV}{RT} \times N_A$$

3. Number Density: The number density $n_d$ (number of molecules per unit volume) can be expressed as:

   $$n_d = \frac{N}{V} = \frac{P}{RT} \times N_A$$

Since the pressure $P$ and temperature $T$ are the same for both balloons, the number density $n_d$ will also be the same for both helium (He) and air.

Thus, the correct answer is:

B) same in both balloons.

Clarification of Incorrect Options

• Option A: more in the He filled balloon: This option is incorrect because, although helium has a lower molar mass than air, the number density depends on pressure and temperature, which are equal for both balloons. Hence, the number of molecules per unit volume is the same.

• Option C: more in air filled balloon: This is also incorrect for the same reason as Option A. The number density is determined by $P$ and $T$, both of which are the same in this scenario.

• Option D: in the ratio of 1:4: This option implies a specific numerical relationship that does not apply. The ratio of number densities is not determined by the type of gas in this context, as the pressure and temperature are constants.

Conclusion

In conclusion, the number of molecules per unit volume in both balloons is the same because they are held at the same pressure and temperature. The Ideal Gas Law confirms that under these conditions, the number density is independent of the type of gas, provided it behaves ideally. Thus, the correct answer is B.