To find the root-mean-square (RMS) velocity of ammonia molecules (NH3) at 45°C, we follow these steps:
1. Determine the molecular mass of NH3:
The molecular mass of ammonia (NH3) is 17.031 g/mol.
2. Convert the molecular mass from grams per mole to kilograms per mole:
Since 1 kilogram equals 1000 grams, we divide by 1000:
[tex]\[
\text{Molecular mass in kg/mol} = \frac{17.031 \text{ g/mol}}{1000} = 0.017031 \text{ kg/mol}
\][/tex]
3. Convert the temperature from Celsius to Kelvin:
To convert Celsius to Kelvin, we add 273.15 to the Celsius temperature:
[tex]\[
\text{Temperature in K} = 45^\circ \text{C} + 273.15 = 318.15 \text{ K}
\][/tex]
4. Use the root-mean-square velocity formula:
The formula for root-mean-square velocity (vrms) is:
[tex]\[
v_{rms} = \sqrt{\frac{3RT}{M}}
\][/tex]
where:
- [tex]\( R \)[/tex] is the universal gas constant, 8.314 J/(mol·K)
- [tex]\( T \)[/tex] is the absolute temperature in Kelvin
- [tex]\( M \)[/tex] is the molar mass in kg/mol
5. Substitute the known values into the formula:
[tex]\[
v_{rms} = \sqrt{\frac{3 \times 8.314 \text{ J/(mol·K)} \times 318.15 \text{ K}}{0.017031 \text{ kg/mol}}}
\][/tex]
6. Calculate the root-mean-square velocity:
Using the given calculations and constants, the root-mean-square velocity of ammonia molecules (NH3) at 45°C is approximately:
[tex]\[
v_{rms} \approx 682.6 \text{ m/s}
\][/tex]
Given the answer choices, the closest to our calculated result is:
[tex]\[
682 \text{ m/s}
\][/tex]
So, the root-mean-square velocity of ammonia molecules (NH3) at 45°C is:
[tex]\[
\boxed{682 \text{ m/s}}
\][/tex]
