Two balloons are filled one with pure He gas and the other with air respectively. If the pressure and temperature of these balloons are same, then the number of molecules per unit volume is

Options

A. More is the He filled balloon
B. same in both balloons
C. more in air filled balloons
D. In the ratio of 1:4

Answer

To solve the problem of determining the number of molecules per unit volume in two balloons filled with helium gas and air, we can use the ideal gas law. Here’s a step-by-step solution:

### Step 1: Understand the Ideal Gas Law
The ideal gas law is given by the equation:
\[ PV = nRT \]
where:
- \( P \) = pressure
- \( V \) = volume
- \( n \) = number of moles of gas
- \( R \) = universal gas constant
- \( T \) = temperature

### Step 2: Relate Moles to Molecules
We can express the number of moles \( n \) in terms of the number of molecules \( N \):
\[ n = \frac{N}{N_A} \]
where \( N_A \) is Avogadro's number (approximately \( 6.022 \times 10^{23} \) molecules/mol).

### Step 3: Substitute for Moles in the Ideal Gas Law
Substituting \( n \) into the ideal gas law gives:
\[ PV = \left(\frac{N}{N_A}\right)RT \]
Rearranging this gives:
\[ N = \frac{P V N_A}{RT} \]

### Step 4: Find Number of Molecules per Unit Volume
To find the number of molecules per unit volume \( n_v \) (which is \( N/V \)), we can rearrange the equation:
\[ n_v = \frac{N}{V} = \frac{P N_A}{RT} \]

### Step 5: Use Boltzmann's Constant
We can express \( R \) in terms of Boltzmann's constant \( k \):
\[ R = N_A k \]
Substituting this into our equation gives:
\[ n_v = \frac{P N_A}{N_A k T} = \frac{P}{k T} \]

### Step 6: Analyze the Given Conditions
Since the problem states that the pressure \( P \) and temperature \( T \) are the same for both balloons, we can conclude that:
\[ n_v \text{ (for He)} = n_v \text{ (for air)} \]

### Conclusion
Thus, the number of molecules per unit volume is the same for both balloons filled with helium and air when the pressure and temperature are equal.

### Final Answer
The number of molecules per unit volume is the same in both balloons.

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