Question

a) What is the root-mean-square speed for a collection of helium-4 atoms at \(300 . \mathrm{K} ?\) b) What is the root-mean-square speed for a collection of helium-3 atoms at \(300 . \mathrm{K} ?\)

Short answer

Question: Calculate the root-mean-square speed of helium-4 atoms and helium-3 atoms at 300 K. Answer: The root-mean-square speed for helium-4 atoms at 300 K is approximately 1.378e3 m/s, and for helium-3 atoms, it is approximately 1.682e3 m/s.

Step-by-step solution

Calculate the mass of helium-4 atoms

To find the mass of helium-4, we need to convert the atomic mass to kilograms. The atomic mass of helium-4 is approximately 4 g/mol. 1 mol contains 6.022e+23 atoms (Avogadro's number) and 1 g = 1e-3 kg. The mass of one helium-4 atom can be calculated as follows: m_He4 = (4 g/mol) * (1e-3 kg/g) / (6.022e+23 atoms/mol) = 6.646e-27 kg.

Calculate the mass of helium-3 atoms

To find the mass of helium-3, we need to convert the atomic mass to kilograms. The atomic mass of helium-3 is approximately 3 g/mol. Use the same conversion as above: m_He3 = (3 g/mol) * (1e-3 kg/g) / (6.022e+23 atoms/mol) = 4.984e-27 kg.

Calculate the root-mean-square speed for helium-4 atoms

Now, we can use the calculated mass of helium-4 atom and the temperature of 300 K to find its root-mean-square speed: \(v_{rms}^{He4} = \sqrt{\frac{3(1.38e-23)(300)}{6.646e-27}}\) \(v_{rms}^{He4} = 1.378e3 \,\text{m/s}\) The root-mean-square speed for helium-4 atoms at 300 K is approximately 1.378e3 m/s.

Calculate the root-mean-square speed for helium-3 atoms

Similarly, we can use the calculated mass of helium-3 atom and the temperature of 300 K to find its root-mean-square speed: \(v_{rms}^{He3} = \sqrt{\frac{3(1.38e-23)(300)}{4.984e-27}}\) \(v_{rms}^{He3} = 1.682e3 \,\text{m/s}\) The root-mean-square speed for helium-3 atoms at 300 K is approximately 1.682e3 m/s.

Key concepts

Kinetic Theory of Gases

At the heart of understanding gaseous behaviors is the kinetic theory of gases, which offers a framework for understanding the motion of gas particles. In essence, this theory posits that gases consist of numerous particles moving in random, straight-line motion, and that the interactions between these particles are entirely elastic collisions, meaning there's no net loss in kinetic energy.

Furthermore, kinetic theory assumes that the average kinetic energy of gas particles is directly proportional to the temperature of the gas in kelvins. It is from this premise that we derive the concept of root-mean-square (rms) speed, which reflects the typical speed of gas particles within a sample. To calculate rms speeds, the formula used incorporates the mass of an individual particle and the temperature of the gas, ensuring that as the temperature increases, so too does the speed of the gas particles.

Atomic Mass Unit Conversion

When delving into problems involving the kinetics of gases, it's fundamental to convert atomic masses from atomic mass units (amu) to kilograms, as the latter is the standard measurement unit in physics. This conversion is key because all formulas related to the kinetic energy or speed of gas particles require mass in kilograms.

The conversion baseline is that 1 amu is approximately equal to 1.66053906660 x 10^-27 kilograms. To determine the mass of one atom in kilograms, if the atomic mass is given in grams per mole, one must also take into account Avogadro's number (6.022 x 10²³ atoms/mol) which tells us the number of particles in one mole of a substance. By combining these conversions, students can tackle complex problems and understand the kinetic behaviors of different gases.

Helium Atoms Kinetics

Let's take a focused look at helium atoms and their kinetics. Helium, with its low atomic number, is often used in these types of exercises due to its simplicity as a noble gas with stable, monoatomic characteristics. Kinetics of helium atoms can be quite revealing in the context of gas behavior and speed calculations.

When comparing the rms speeds of helium-3 and helium-4 atoms at the same temperature, the lighter helium-3 atoms move faster. This practical illustration shines a spotlight on another kinetic theory tenet: the inverse relationship between mass and speed at a constant temperature. It demonstrates that as the mass of atoms decreases, their speed, given sufficient thermal energy, increases, and this dynamic directly influences phenomena such as diffusion rates, sound speed in gases, and even the efficiency of gas-based equipment.