The masses of 1 mole of various gases are as follows: hydrogen about 2 grams, helium about 4 grams, nitrogen about 28 grams, oxygen about 32 grams and carbon dioxide about 44 grams. On the average how fast does a molecule of each gas move at 333 Celsius?
To calculate the average speed of a gas molecule at a given temperature, we can use the root mean square speed formula:
v = sqrt((3kT) / (m))
where:
- v is the average speed of the gas molecule,
- k is the Boltzmann constant (1.38 * 10^-23 J/K),
- T is the temperature in Kelvin, and
- m is the molar mass of the gas molecule.
Let's calculate the average speed of each gas molecule at a temperature of 333 degrees Celsius (606 Kelvin):
1. Hydrogen:
m = 2 grams/mol = 0.002 kg/mol
T = 333 + 273 = 606 K
v = sqrt((3 * 1.38 * 10^-23 J/K * 606 K) / (0.002 kg/mol))
≈ 1.95 * 10^3 m/s
2. Helium:
m = 4 grams/mol = 0.004 kg/mol
T = 333 + 273 = 606 K
v = sqrt((3 * 1.38 * 10^-23 J/K * 606 K) / (0.004 kg/mol))
≈ 1.38 * 10^3 m/s
3. Nitrogen:
m = 28 grams/mol = 0.028 kg/mol
T = 333 + 273 = 606 K
v = sqrt((3 * 1.38 * 10^-23 J/K * 606 K) / (0.028 kg/mol))
≈ 5.56 * 10^2 m/s
4. Oxygen:
m = 32 grams/mol = 0.032 kg/mol
T = 333 + 273 = 606 K
v = sqrt((3 * 1.38 * 10^-23 J/K * 606 K) / (0.032 kg/mol))
≈ 5.09 * 10^2 m/s
5. Carbon dioxide:
m = 44 grams/mol = 0.044 kg/mol
T = 333 + 273 = 606 K
v = sqrt((3 * 1.38 * 10^-23 J/K * 606 K) / (0.044 kg/mol))
≈ 4.62 * 10^2 m/s
Therefore, at a temperature of 333 degrees Celsius, on average:
- A hydrogen molecule moves at a speed of approximately 1.95 * 10^3 m/s.
- A helium molecule moves at a speed of approximately 1.38 * 10^3 m/s.
- A nitrogen molecule moves at a speed of approximately 5.56 * 10^2 m/s.
- An oxygen molecule moves at a speed of approximately 5.09 * 10^2 m/s.
- A carbon dioxide molecule moves at a speed of approximately 4.62 * 10^2 m/s.
To calculate the average speed of a gas molecule at a certain temperature, we can use the root mean square (RMS) speed equation. The RMS speed of a gas molecule can be determined using the formula:
v = √(3RT / M)
Where:
v = RMS speed of a gas molecule
R = Universal gas constant (8.314 J/(mol·K))
T = Temperature in Kelvin (Celsius + 273.15)
M = Molar mass of the gas in kilograms (To convert grams to kilograms, divide by 1000)
Let's plug in the values for each gas:
For Hydrogen:
Molar mass (H2) = 2g/mol
T = 333°C + 273.15 = 606.15 K
M = 2g/mol / 1000 = 0.002 kg/mol
vhydrogen = √(3 * 8.314 J/(mol·K) * 606.15 K / 0.002 kg/mol)
For Helium:
Molar mass (He) = 4g/mol
T = 333°C + 273.15 = 606.15 K
M = 4g/mol / 1000 = 0.004 kg/mol
vhelium = √(3 * 8.314 J/(mol·K) * 606.15 K / 0.004 kg/mol)
For Nitrogen:
Molar mass (N2) = 28g/mol
T = 333°C + 273.15 = 606.15 K
M = 28g/mol / 1000 = 0.028 kg/mol
vnitrogen = √(3 * 8.314 J/(mol·K) * 606.15 K / 0.028 kg/mol)
For Oxygen:
Molar mass (O2) = 32g/mol
T = 333°C + 273.15 = 606.15 K
M = 32g/mol / 1000 = 0.032 kg/mol
voxygen = √(3 * 8.314 J/(mol·K) * 606.15 K / 0.032 kg/mol)
For Carbon Dioxide:
Molar mass (CO2) = 44g/mol
T = 333°C + 273.15 = 606.15 K
M = 44g/mol / 1000 = 0.044 kg/mol
vco2 = √(3 * 8.314 J/(mol·K) * 606.15 K / 0.044 kg/mol)
By substituting the values and calculating the equations, we can find the average speed of each gas molecule at 333°C.