Find rms speed of Aluminium molecules at temperature `27^(@)C`.

Options

A. 526m/s
B. 530m/s
C. 528m/s
D. 555m/s

Explanation

To find the root mean square (RMS) speed of aluminum molecules at a temperature of 27°C, we can follow these steps:

### Step 1: Convert Temperature to Kelvin
The temperature in Celsius needs to be converted to Kelvin using the formula:
\[ T(K) = T(°C) + 273 \]
Given \( T = 27°C \):
\[ T = 27 + 273 = 300 \, K \]

### Step 2: Identify the Constants
We need the following constants for the calculation:
- Gas constant \( R = 8.31 \, \text{J K}^{-1} \text{mol}^{-1} \)
- Atomic mass of aluminum \( m = 27 \, \text{g/mol} \)

### Step 3: Convert Atomic Mass to Kilograms
Since we need the mass in kilograms for the RMS speed formula, we convert grams to kilograms:
\[ m = 27 \, \text{g/mol} = 27 \times 10^{-3} \, \text{kg/mol} \]

### Step 4: Use the RMS Speed Formula
The formula for calculating the RMS speed \( v_{rms} \) is:
\[ v_{rms} = \sqrt{\frac{3RT}{M}} \]
Where:
- \( R \) is the gas constant,
- \( T \) is the temperature in Kelvin,
- \( M \) is the molar mass in kg/mol.

### Step 5: Substitute the Values into the Formula
Substituting the values into the formula:
\[ v_{rms} = \sqrt{\frac{3 \times 8.31 \, \text{J K}^{-1} \text{mol}^{-1} \times 300 \, K}{27 \times 10^{-3} \, \text{kg/mol}}} \]

### Step 6: Calculate the RMS Speed
Now, we calculate the value inside the square root:
1. Calculate the numerator:
\[ 3 \times 8.31 \times 300 = 7479 \]
2. Calculate the denominator:
\[ 27 \times 10^{-3} = 0.027 \]
3. Now, divide the numerator by the denominator:
\[ \frac{7479}{0.027} \approx 276,555.56 \]
4. Finally, take the square root:
\[ v_{rms} \approx \sqrt{276555.56} \approx 526.30 \, \text{m/s} \]

### Step 7: Round the Answer
Rounding the answer gives:
\[ v_{rms} \approx 526 \, \text{m/s} \]

### Final Answer
The RMS speed of aluminum molecules at 27°C is approximately **526 m/s**.

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