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

Options

A. 432m/s
B. 532m/s
C. 632m/s
D. 482m/s

Explanation

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

### Step 1: Convert the temperature from Celsius to Kelvin
The formula to convert Celsius to Kelvin is:
\[ T(K) = T(°C) + 273 \]

Given:
\[ T(°C) = 27 \]
So,
\[ T(K) = 27 + 273 = 300 \, K \]

### Step 2: Identify the atomic mass of calcium
The atomic mass of calcium (Ca) is approximately 40 g/mol. To use it in the rms speed formula, we need to convert it to kilograms:
\[ m = 40 \, g/mol = 40 \times 10^{-3} \, kg/mol = 0.040 \, kg/mol \]

### Step 3: Use the rms speed formula
The formula for the root mean square speed (v_rms) is:
\[ v_{rms} = \sqrt{\frac{3RT}{m}} \]
Where:
- \( R \) is the gas constant, \( R = 8.31 \, J/(K \cdot mol) \)
- \( T \) is the temperature in Kelvin
- \( m \) is the molar mass in kg/mol

### Step 4: Substitute the values into the formula
Now we can substitute the values we have into the formula:
\[ v_{rms} = \sqrt{\frac{3 \times 8.31 \, J/(K \cdot mol) \times 300 \, K}{0.040 \, kg/mol}} \]

### Step 5: Calculate the value inside the square root
First, calculate the numerator:
\[ 3 \times 8.31 \times 300 = 7473 \, J/mol \]

Now, calculate the entire expression:
\[ v_{rms} = \sqrt{\frac{7473}{0.040}} \]

Calculating the division:
\[ \frac{7473}{0.040} = 186825 \]

### Step 6: Take the square root
Now, take the square root:
\[ v_{rms} = \sqrt{186825} \approx 432.0 \, m/s \]

### Final Answer
The root mean square speed of calcium molecules at 27°C is approximately:
\[ v_{rms} \approx 432 \, m/s \]

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