Understanding the Speed of Helium Atoms in a Balloon at Room Temperature

A boy wonders about the speed of helium atoms inside a balloon. Let's explore how we can calculate the average speed of these gas particles using the principles of the kinetic theory of gases.

Theoretical Background

The kinetic theory of gases provides a fundamental framework for understanding the behavior of gas particles. According to this theory, gas molecules move randomly and collide with each other and the walls of their container. The speed of gas particles is influenced by the temperature and the mass of the particles.

Calculating the Average Speed of Helium Atoms

We can use the following equation to estimate the average speed of gas particles:

[ v sqrt{frac{3kT}{m}} ]

Where:

(k) is the Boltzmann constant (1.38 times 10^{-23} text{J/K}) (T) is the absolute temperature in Kelvin (m) is the mass of a helium atom in kilograms

Converting the Temperature

The room temperature is given as 20 degrees Celsius. To use this in our equation, we need to convert it to Kelvin:

[ T 20 text{°C} 273.15 293.15 text{K} ]

Finding the Mass of a Helium Atom

The molar mass of helium is approximately 4 grams per mole. To convert this to kilograms, we use Avogadro's number (6.022 times 10^{23} text{atoms/mol}):

[ m frac{4 text{g/mol}}{1000} times frac{1 text{mol}}{6.022 times 10^{23} text{atoms/mol}} approx 6.64 times 10^{-27} text{kg} ]

Calculating the Average Speed

Now, we can plug in the values into the equation to find the average speed of the helium atoms:

[ v sqrt{frac{3 times 1.38 times 10^{-23} text{J/K} times 293.15 text{K}}{6.64 times 10^{-27} text{kg}}} ]

First, let's calculate the numerator:

[ 3 times 1.38 times 10^{-23} times 293.15 approx 1.21 times 10^{-20} text{J kg} ]

Now, divide by the mass:

[ frac{1.21 times 10^{-20}}{6.64 times 10^{-27}} approx 1.82 times 10^{6} text{m}^2/text{s}^2 ]

Finally, take the square root:

[ v approx sqrt{1.82 times 10^{6}} approx 1340 text{m/s} ]

Therefore, the average speed of the helium atoms in the balloon at 20 degrees Celsius is approximately 1340 meters per second.

Additional Insights

The concept of average speed in the kinetic theory of gases does not exactly match the traditional idea of orbital motion, as suggested by the Bohr model. Instead, it describes the statistical motion of particles, which can be thought of as a three-dimensional pendulum with a range of speeds and momenta.

The Planck's equation, which involves position and momentum, further clarifies the distribution of speeds among gas particles. This distribution is such that the majority of helium particles move at an average speed, with a few reaching a maximum speed due to collisions and energy exchange.

It is important to note that the balloon itself is in a state of equilibrium, and the helium atoms within the balloon follow the principles of the kinetic theory of gases under the given temperature conditions.

Key Concepts: Helium atoms, gas particle speed, kinetic theory of gases