What is the rms speed of N2 molecules at 281 K?
What is the rms speed of He atoms at 281 K?
where R=8.314
v_rms = sqrt(3RT/M) where M is the 28 for N2, and 4 for He
To find the root mean square (rms) speed of gas molecules or atoms, you can use the following formula:
v = √(3RT/M)
Here, v represents the rms speed of the particles, R is the gas constant (8.314 J/(mol·K)), T is the temperature in Kelvin, and M is the molar mass of the gas in kilograms per mole.
For the first question, we are dealing with nitrogen gas (N2) molecules at a temperature of 281 K. The molar mass of N2 is 28 g/mol, which is equivalent to 0.028 kg/mol.
Plugging the values into the formula, we have:
v = √(3 × 8.314 J/(mol·K) × 281 K / 0.028 kg/mol)
Calculating the expression within the square root, we get:
v = √(3 × 8.314 × 281 / 0.028) m/s
Simplifying further:
v = √(>3 × 8.314 × 281 × 0.028) m/s
v ≈ √(299.132) m/s
v ≈ 17.29 m/s (rounded to two decimal places)
Therefore, the rms speed of N2 molecules at a temperature of 281 K is approximately 17.29 m/s.
Similarly, for the second question, we are dealing with helium (He) atoms at the same temperature of 281 K. The molar mass of helium is 4 g/mol, which is equivalent to 0.004 kg/mol.
Using the same formula mentioned above, we can calculate the rms speed:
v = √(3 × 8.314 J/(mol·K) × 281 K / 0.004 kg/mol)
Calculating the expression within the square root, we get:
v = √(3 × 8.314 × 281 / 0.004) m/s
v ≈ √(56.342) m/s
v ≈ 7.51 m/s (rounded to two decimal places)
Therefore, the rms speed of helium atoms at a temperature of 281 K is approximately 7.51 m/s.