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

Options

A. 100m/s
B. 1034m/s
C. 1000m/s
D. 1459m/s

Explanation

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

### Step 1: Convert Temperature to Kelvin
The temperature given is 27°C. To convert this to Kelvin, we use the formula:
\[ T(K) = T(°C) + 273 \]
Substituting the value:
\[ T = 27 + 273 = 300 \, K \]

### Step 2: Identify the Molar Mass of Lithium
The molar mass of lithium (Li) is approximately 7 g/mol. To use this in our calculations, we need to convert it to kilograms per mole:
\[ M = 7 \, g/mol = 7 \times 10^{-3} \, kg/mol \]

### Step 3: Use the RMS Speed Formula
The formula for calculating the rms speed (\(v_{rms}\)) is given by:
\[ v_{rms} = \sqrt{\frac{3RT}{M}} \]
where:
- \(R\) is the gas constant, \(8.31 \, J/(K \cdot mol)\)
- \(T\) is the temperature in Kelvin
- \(M\) is the molar mass in kg/mol

### Step 4: Substitute Values into the Formula
Now we substitute the values into the rms speed formula:
\[ v_{rms} = \sqrt{\frac{3 \times 8.31 \, J/(K \cdot mol) \times 300 \, K}{7 \times 10^{-3} \, kg/mol}} \]

### Step 5: Calculate the Value
Calculating the numerator:
\[ 3 \times 8.31 \times 300 = 7479 \, J/mol \]
Now, substituting this into the formula:
\[ v_{rms} = \sqrt{\frac{7479}{7 \times 10^{-3}}} \]
Calculating the denominator:
\[ 7 \times 10^{-3} = 0.007 \, kg/mol \]
Now we can calculate:
\[ v_{rms} = \sqrt{1067000} \approx 1033.6 \, m/s \]

### Final Answer
Thus, the rms speed of lithium molecules at 27°C is approximately:
\[ v_{rms} \approx 1034 \, m/s \]