7. What is the rms speed of nitrogen molecules contained in an 8.5 m3 volume at 3.1 atm if the total amount of nitrogen is 1800 mol?
To find the root mean square (rms) speed of the nitrogen molecules, we can use the ideal gas law and the formula for rms speed.
Step 1: Convert the volume to liters.
We are given the volume as 8.5 m^3. To convert this to liters, we multiply by the conversion factor:
1 m^3 = 1000 liters
So, 8.5 m^3 * 1000 liters/m^3 = 8500 liters.
Step 2: Convert the pressure to Pascals.
The pressure is given as 3.1 atm. To convert this to Pascals, we multiply by the conversion factor:
1 atm = 101325 Pascals
So, 3.1 atm * 101325 Pascals/atm = 314,175 Pascals.
Step 3: Calculate the number of nitrogen molecules.
The total amount of nitrogen is given as 1800 mol. Since 1 mole contains Avogadro's number (6.022 × 10^23) of molecules, we multiply the number of moles by Avogadro's number to get the total number of molecules:
1800 mol * 6.022 × 10^23 molecules/mol = 1.0836 × 10^27 molecules.
Step 4: Calculate the rms speed.
The formula for rms speed is given by:
v_rms = sqrt((3 * R * T) / M)
where R is the ideal gas constant, T is the temperature in Kelvin, and M is the molar mass of the gas.
The molar mass of nitrogen (N2) is 28 g/mol.
Step 5: Calculate the temperature.
Since the ideal gas law is PV = nRT, we can rearrange it to solve for T:
T = (P * V) / (n * R)
where P is the pressure in Pascals, V is the volume in cubic meters, n is the number of moles, and R is the ideal gas constant (8.314 J/(mol·K)).
Using the given values:
T = (314175 Pascals * 8.5 m^3) / (1800 mol * 8.314 J/(mol·K)).
Step 6: Calculate the rms speed.
Now, we can substitute the values into the rms speed formula:
v_rms = sqrt((3 * 8.314 J/(mol·K) * T) / (28 g/mol)).
By plugging in the calculated value of T into the formula, we can find the rms speed of the nitrogen molecules.