1. Convert Units
* Temperature: 30°C = 303.15 K (add 273.15 to convert from Celsius to Kelvin)
* Pressure: 2.00 atm = 2.03 x 10^5 Pa (1 atm = 1.01325 x 10^5 Pa)
* Mass: 4 u = 6.64 x 10^-27 kg (1 u = 1.66054 x 10^-27 kg)
2. Use the Ideal Gas Law
The ideal gas law relates pressure (P), volume (V), number of moles (n), the ideal gas constant (R), and temperature (T):
PV = nRT
We can use this to find the volume:
V = (nRT)/P
3. Calculate the rms Speed
The root-mean-squared speed (v_rms) of an ideal gas is given by:
v_rms = √(3RT/M)
Where:
* R is the ideal gas constant (8.314 J/(mol·K))
* T is the temperature in Kelvin
* M is the molar mass of the gas in kg/mol (M = 4 g/mol = 0.004 kg/mol for Helium)
Calculations
1. Find the volume:
V = (1 mol * 8.314 J/(mol·K) * 303.15 K) / (2.03 x 10^5 Pa)
V ≈ 0.0124 m³
2. Calculate the rms speed:
v_rms = √(3 * 8.314 J/(mol·K) * 303.15 K / 0.004 kg/mol)
v_rms ≈ 1360 m/s
Therefore, the root-mean-squared speed of helium atoms in one mole of an ideal gas at a pressure of 2.00 atmospheres and a temperature of 30 degrees Celsius is approximately 1360 m/s.
