A bubble of air is underwater at temperature `15^(@)C` and the pressure `1.5` bar. If the bubble rises to the surface where the temperature is `25^(@)C` and the pressure is `1.0` bar, what will happen to the volume of the bubble?

Options

A. (a) Volume will become greater by a factor of 1.6
B. (b) Volume will become greater by a factor of 1.1
C. (c) Volume will become smaller by a factor of 0.70
D. (d) Volume will become greater by a factor of 2.5.

Explanation

To solve the problem of how the volume of a bubble changes as it rises from underwater to the surface, we can use the ideal gas law and the relationship between pressure, volume, and temperature. Here’s a step-by-step breakdown of the solution:

### Step 1: Identify the Given Values
- Initial temperature (T1) = 15°C
- Initial pressure (P1) = 1.5 bar
- Final temperature (T2) = 25°C
- Final pressure (P2) = 1.0 bar

### Step 2: Convert Temperatures to Kelvin
To use the ideal gas law, we need to convert the temperatures from Celsius to Kelvin using the formula:
\[ T(K) = T(°C) + 273 \]

- For T1:
\[ T1 = 15 + 273 = 288 \, K \]

- For T2:
\[ T2 = 25 + 273 = 298 \, K \]

### Step 3: Use the Ideal Gas Law Relationship
According to the ideal gas law, for a given amount of gas, the relationship between pressure, volume, and temperature can be expressed as:
\[ \frac{P1 \cdot V1}{T1} = \frac{P2 \cdot V2}{T2} \]

### Step 4: Rearrange the Equation
We want to find the relationship between the initial volume (V1) and the final volume (V2). Rearranging the equation gives us:
\[ \frac{V1}{V2} = \frac{P2 \cdot T1}{P1 \cdot T2} \]

### Step 5: Substitute the Known Values
Now, we can substitute the known values into the equation:
- P1 = 1.5 bar
- P2 = 1.0 bar
- T1 = 288 K
- T2 = 298 K

Thus, we have:
\[ \frac{V1}{V2} = \frac{1.0 \cdot 288}{1.5 \cdot 298} \]

### Step 6: Calculate the Ratio
Now, we can calculate the ratio:
\[ \frac{V1}{V2} = \frac{288}{1.5 \cdot 298} \]
\[ = \frac{288}{447} \approx 0.645 \]

### Step 7: Find V2 in Terms of V1
To find V2 in terms of V1, we take the reciprocal of the ratio:
\[ \frac{V2}{V1} = \frac{1}{0.645} \approx 1.55 \]

### Step 8: Conclusion
This means that the final volume (V2) is approximately 1.55 times the initial volume (V1). Therefore, the volume of the bubble increases as it rises to the surface.

### Final Answer
The volume of the bubble will increase, and specifically, it will be approximately 1.55 times the initial volume.

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