Find the rms speed of H2O molecules in atmospheric water vapor at a temperature of 13.2° C.
To find the root mean square (rms) speed of water molecules in atmospheric water vapor, we can use the following formula:
v(rms) = sqrt((3 * k * T) / m)
where:
v(rms) is the root mean square speed of the molecules,
k is Boltzmann's constant (1.38 x 10^-23 J/K),
T is the temperature in Kelvin,
m is the molar mass of the molecule in kg.
To solve this problem, we need to follow a few steps:
Step 1: Convert the temperature from Celsius to Kelvin:
T(K) = T(C) + 273.15
Given T(C) = 13.2 °C, we can convert it into Kelvin:
T(K) = 13.2 + 273.15
Step 2: Determine the molar mass of water (H2O):
In the case of water (H2O), the molar mass is calculated as follows:
Molar mass of H2O = (2 * Molar mass of H) + Molar mass of O
The molar mass of H is approximately 1 g/mol, and the molar mass of O is approximately 16 g/mol. Therefore,
Molar mass of H2O = (2 * 1 g/mol) + 16 g/mol
Molar mass of H2O = 18 g/mol
Step 3: Convert the molar mass of water from grams to kilograms:
Molar mass of H2O = 18 g/mol = 0.018 kg/mol
Step 4: Substitute the values into the formula:
v(rms) = sqrt((3 * k * T) / m)
v(rms) = sqrt((3 * 1.38 x 10^-23 J/K) * (13.2 + 273.15) K) / 0.018 kg/mol
Step 5: Calculate the rms speed:
v(rms) = sqrt((3 * 1.38 x 10^-23 J/K) * (286.35 K)) / 0.018 kg/mol
Now, plug the values into a calculator to find the rms speed of H2O molecules in atmospheric water vapor at a temperature of 13.2 °C.
Note: Remember to convert the final answer back to the desired units if needed.