Find the difference of pressure between inside and outside of a soap bubble 5mm in diameter . Surface tension of soap solution is `3.2 xx 10^(-2) Nm^(-1)`

Explanation

To find the difference of pressure between the inside and outside of a soap bubble, we can use the formula for the excess pressure in a soap bubble, which is given by:

\[ \Delta P = \frac{4 \sigma}{R} \]

where:
- \(\Delta P\) is the difference of pressure,
- \(\sigma\) is the surface tension of the soap solution,
- \(R\) is the radius of the bubble.

### Step-by-Step Solution:

1. **Identify the Given Values**:
- Diameter of the soap bubble, \(D = 5 \, \text{mm} = 5 \times 10^{-3} \, \text{m}\)
- Surface tension of soap solution, \(\sigma = 3.2 \times 10^{-2} \, \text{N/m}\)

2. **Calculate the Radius**:
- The radius \(R\) is half of the diameter:
\[ R = \frac{D}{2} = \frac{5 \times 10^{-3}}{2} = 2.5 \times 10^{-3} \, \text{m} \]

3. **Substitute the Values into the Formula**:
- Now substitute \(\sigma\) and \(R\) into the formula for \(\Delta P\):
\[ \Delta P = \frac{4 \sigma}{R} = \frac{4 \times (3.2 \times 10^{-2})}{2.5 \times 10^{-3}} \]

4. **Calculate the Numerator**:
- Calculate \(4 \times 3.2 \times 10^{-2}\):
\[ 4 \times 3.2 = 12.8 \quad \Rightarrow \quad 12.8 \times 10^{-2} \]

5. **Calculate the Difference of Pressure**:
- Now divide by the radius:
\[ \Delta P = \frac{12.8 \times 10^{-2}}{2.5 \times 10^{-3}} = \frac{12.8}{2.5} \times 10^{1} = 5.12 \times 10^{1} \, \text{N/m}^2 \]
- Simplifying further:
\[ \Delta P = 51.2 \, \text{N/m}^2 \]

6. **Final Result**:
- The difference of pressure between the inside and outside of the soap bubble is:
\[ \Delta P \approx 51.2 \, \text{N/m}^2 \]