calculate the rms speeds of He at 10, 100, and 1000 K in meters per second and in miles per hour. What values would be otained if the pressure was 10^-10 bar
To calculate the root mean square (RMS) speed of a gas, you can use the following formula:
v_rms = √((3kT) / m)
Where:
- v_rms is the RMS speed
- k is the Boltzmann constant (1.38 × 10^-23 J/K)
- T is the temperature in Kelvin
- m is the molar mass of the gas in kilograms per mole
First, let's calculate the RMS speeds of helium (He) at 10 K, 100 K, and 1000 K:
1. Convert the temperature from Kelvin to Celsius:
T (in °C) = T (in K) - 273.15
2. Calculate the molar mass:
The molar mass of helium (He) is approximately 4.002602 g/mol.
Convert it to kilograms per mole:
m (in kg/mol) = 4.002602 × 10^-3 kg/mol
3. Substitute the values into the RMS speed formula:
v_rms = √((3 × 1.38 × 10^-23 J/K × T) / (4.002602 × 10^-3 kg/mol))
To convert the RMS speeds from meters per second to miles per hour, use the following conversion factor:
1 m/s ≈ 2.237 miles/hour
Now, let's calculate the RMS speeds of helium (He) at different temperatures:
1. At 10 K:
- Convert temperature from Kelvin to Celsius: 10 K - 273.15 = -263.15 °C
- Substitute the values into the RMS speed formula:
v_rms = √((3 × 1.38 × 10^-23 J/K × 10 K) / (4.002602 × 10^-3 kg/mol))
2. At 100 K:
- Convert temperature from Kelvin to Celsius: 100 K - 273.15 = -173.15 °C
- Substitute the values into the RMS speed formula:
v_rms = √((3 × 1.38 × 10^-23 J/K × 100 K) / (4.002602 × 10^-3 kg/mol))
3. At 1000 K:
- Convert temperature from Kelvin to Celsius: 1000 K - 273.15 = 726.85 °C
- Substitute the values into the RMS speed formula:
v_rms = √((3 × 1.38 × 10^-23 J/K × 1000 K) / (4.002602 × 10^-3 kg/mol))
If the pressure is 10^-10 bar, it does not affect the calculation of RMS speeds because the RMS speed is solely dependent on temperature and molar mass.