Two glass bulbs A and B are connected by a very small tube having a stop cock. Bulb A has a volume of 100 `cm^(3)` and contained the gas, while bulb B was empty. On opening th stop cock. The pressure fell down to 40%. The volume of the bulb B must be:

Options

A. `100 cm^(3)`
B. `200 cm^(3)`
C. `250 cm^(3)`
D. `400 cm^(3)`

Explanation

To solve the problem, we need to apply the ideal gas law and the concept of pressure-volume relationships. Here’s a step-by-step solution:

### Step 1: Understand the Setup
We have two glass bulbs, A and B. Bulb A has a volume of 100 cm³ and contains gas, while bulb B is empty. When the stopcock is opened, the gas from bulb A will expand into bulb B.

### Step 2: Apply the Ideal Gas Law
According to the ideal gas law, for a given amount of gas at constant temperature, the product of pressure and volume is constant:
\[ P_1 V_1 = P_2 V_2 \]

Where:
- \( P_1 \) = initial pressure in bulb A
- \( V_1 \) = volume of bulb A (100 cm³)
- \( P_2 \) = final pressure after opening the stopcock (40% of \( P_1 \))
- \( V_2 \) = volume of bulb B (which we need to find)

### Step 3: Express the Pressures
Let’s denote the initial pressure in bulb A as \( P_1 = k \). Then, the final pressure \( P_2 \) after the stopcock is opened is:
\[ P_2 = 0.4k \]

### Step 4: Set Up the Equation
Substituting the known values into the equation:
\[ P_1 V_1 = P_2 (V_1 + V_2) \]
This becomes:
\[ k \cdot 100 = 0.4k \cdot (100 + V_2) \]

### Step 5: Simplify the Equation
We can cancel \( k \) from both sides (assuming \( k \neq 0 \)):
\[ 100 = 0.4(100 + V_2) \]

### Step 6: Distribute and Rearrange
Distributing \( 0.4 \):
\[ 100 = 40 + 0.4V_2 \]
Now, subtract 40 from both sides:
\[ 60 = 0.4V_2 \]

### Step 7: Solve for \( V_2 \)
To find \( V_2 \), divide both sides by 0.4:
\[ V_2 = \frac{60}{0.4} = 150 \text{ cm}^3 \]

### Step 8: Calculate Total Volume
The total volume when the gas expands into bulb B is:
\[ V_{total} = V_1 + V_2 = 100 + 150 = 250 \text{ cm}^3 \]

### Final Answer
The volume of bulb B must be:
\[ V_B = 150 \text{ cm}^3 \]