If He(g) has an average kinetic energy of 4190 J/mol under certain conditions, what is the root mean square speed of Cl2(g) molecules under the same conditions?
To find the root mean square (rms) speed of Cl2(g) molecules, we can use the equation:
v_rms = √(3RT / M)
Where:
- v_rms is the root mean square speed
- R is the ideal gas constant (8.314 J/mol·K)
- T is the temperature in Kelvin
- M is the molar mass of Cl2(g) in kg/mol
First, let's find the molar mass of Cl2(g):
The molar mass of Cl2 is 35.5 g/mol * 2 = 71 g/mol.
Converting grams to kilograms: M = 71 g/mol / 1000 g/kg = 0.071 kg/mol
Now, let's assume the temperature is given or provided. If it is not given, we won't be able to calculate the rms speed.
Assuming T = 298 K, let's calculate the rms speed:
v_rms = √(3RT / M)
v_rms = √(3 * 8.314 J/mol·K * 298 K / 0.071 kg/mol)
v_rms = √((3 * 8.314 J * 298 K) / (0.071 kg))
v_rms = √(7416.17 J / 0.071 kg)
v_rms = √(104366.479 J/kg)
Using a calculator, we find that v_rms ≈ 323.05 m/s.
Therefore, the root mean square speed of Cl2(g) molecules under the given conditions is approximately 323.05 m/s.
To find the root mean square (rms) speed of Cl2(g) molecules, we can use the relationship between average kinetic energy and rms speed.
The average kinetic energy (KE_avg) of a gas molecule is given by the equation:
KE_avg = (3/2) * k * T
Where:
- KE_avg is the average kinetic energy
- k is the Boltzmann constant (1.38064852 × 10^-23 J/K)
- T is the temperature in Kelvin
Since the average kinetic energy of He(g) molecules is given as 4190 J/mol, we can set up the following equation:
(3/2) * k * T_he = 4190 J/mol
To find the rms speed of Cl2(g) molecules, we need to related the average kinetic energy of Cl2(g) to that of He(g). The relationship between rms speed (v) and average kinetic energy (KE_avg) is given by the equation:
KE_avg = (1/2) * m * v^2
Where:
- KE_avg is the average kinetic energy
- m is the molar mass of the gas (in kg/mol)
- v is the rms speed
Since the molar mass of Cl2 is 71 g/mol, we will convert it to kg/mol by dividing by 1000:
m = 71 g/mol / 1000 g/kg = 0.071 kg/mol
We can substitute this value into the equation:
(1/2) * m * v_cl2^2 = (3/2) * k * T_he
Simplifying the equation:
v_cl2^2 = (3/2) * (k * T_he) / m
Finally, to find the rms speed, we take the square root of both sides:
v_cl2 = √[(3/2) * (k * T_he) / m]
Now, we can substitute the given values into the equation to calculate the rms speed of Cl2(g) molecules.