a 2 l gas bulb contains 4.07*10^23 N2 molecules. If the pressure is 3.05bar, what is the rms speed of the N2 molecules? What is the temperature?
To find the root-mean-square (rms) speed of the N2 molecules and the temperature in the gas bulb, we can apply the ideal gas law and the kinetic theory of gases. Here are the steps to find the answers:
1. Convert the pressure from bar to pascal:
1 bar = 100,000 pascal.
Therefore, 3.05 bar = 3.05 × 100,000 = 305,000 pascal.
2. Determine the number of moles of N2 molecules:
The ideal gas law equation is given by: PV = nRT.
Where:
P = pressure in pascal
V = volume of the gas bulb in liters (we are not given the volume, but it is not required to solve for the rms speed and temperature)
n = number of moles of gas
R = ideal gas constant (8.314 J/mol·K)
T = temperature in Kelvin
Rearrange the equation to solve for n:
n = PV / RT.
The volume (V) will cancel out, so you don't need it for this calculation.
Given:
P = 305,000 pascal
R = 8.314 J/mol·K
T = unknown
To find n, we need to know the volume (V) of the gas bulb. However, since the volume is not given, we cannot calculate the exact number of moles. But, we can still provide an answer in terms of the rms speed and temperature.
3. Calculate the rms speed of the N2 molecules:
The formula for rms speed of a gas is given by:
vrms = sqrt((3RT) / (molar mass))
Where:
R = ideal gas constant (8.314 J/mol·K)
T = temperature in Kelvin
molar mass = mass of one mole of N2 molecules (28 g/mol)
We already found that n is indeterminate without knowing V, but we can still present the answer in terms of temperature.
4. Find the temperature:
Rearrange the rms speed equation to solve for T:
T = (vrms^2 * molar mass) / (3R)
Substituting the given values:
T = (vrms^2 * 28) / (3 * 8.314)
Now we can plug in the values to calculate the temperature.
5. Calculate the rms speed of the N2 molecules:
Substitute the known values into the rms speed equation:
vrms = sqrt((3RT) / (molar mass))
T is the temperature we just calculated, so we'll substitute T into the equation to find vrms.
By following these steps, you can calculate the rms speed and temperature for the given conditions.