Calculate the rms speed of smoke particles of mass `5 xx 10^(-17) kg` in their Brownian motion in air at NTP. Given `k_(B) = 1.38 xx 10^(-23) J//K`

Options

A. `3xx10^(-2)ms^(-1)`
B. `1.5xx10^(-2)ms^(-1)`
C. `3xx10^(-3)ms^(-1)`
D. `1.5xx10^(-3)ms^(-1)`

Explanation

To calculate the root mean square (rms) speed of smoke particles in Brownian motion, we can use the formula:

\[ v_{rms} = \sqrt{\frac{3k_B T}{m}} \]

where:
- \( v_{rms} \) is the root mean square speed,
- \( k_B \) is the Boltzmann constant,
- \( T \) is the absolute temperature in Kelvin,
- \( m \) is the mass of the particle.

### Step 1: Write down the given data
- Mass of the particle, \( m = 5 \times 10^{-17} \, \text{kg} \)
- Temperature at NTP, \( T = 273 \, \text{K} \)
- Boltzmann constant, \( k_B = 1.38 \times 10^{-23} \, \text{J/K} \)

### Step 2: Substitute the values into the rms speed formula
Now, we can substitute the values into the formula:

\[ v_{rms} = \sqrt{\frac{3 \times (1.38 \times 10^{-23} \, \text{J/K}) \times (273 \, \text{K})}{5 \times 10^{-17} \, \text{kg}}} \]

### Step 3: Calculate the numerator
First, calculate the numerator:

\[ 3 \times (1.38 \times 10^{-23}) \times (273) = 1.13154 \times 10^{-20} \, \text{J} \]

### Step 4: Calculate the denominator
Now, the denominator is simply:

\[ 5 \times 10^{-17} \, \text{kg} \]

### Step 5: Divide the numerator by the denominator
Now, we divide the numerator by the denominator:

\[ \frac{1.13154 \times 10^{-20}}{5 \times 10^{-17}} = 2.26308 \times 10^{-4} \, \text{m}^2/\text{s}^2 \]

### Step 6: Take the square root
Finally, we take the square root to find the rms speed:

\[ v_{rms} = \sqrt{2.26308 \times 10^{-4}} \approx 1.5 \times 10^{-2} \, \text{m/s} \]

### Conclusion
Thus, the rms speed of the smoke particles is:

\[ v_{rms} \approx 1.5 \times 10^{-2} \, \text{m/s} \]