The escape velocity of planet earth is 11.2 km/s. At what temperature would helium atoms be able to escape the earth's gravitational pull? (Note: you're calculating the average speed here, which means roughly 50% of the atoms would escape at any given time)
To calculate the temperature at which helium atoms would be able to escape the Earth's gravitational pull, we can use the formula for the average speed of gas molecules:
v_avg = √(8*k*T / (π*m))
Where:
v_avg is the average speed of the gas molecules,
k is the Boltzmann constant (1.38 x 10^-23 J/K),
T is the temperature in Kelvin,
m is the mass of individual helium atom (4 atomic mass units or 6.64 x 10^-27 kg).
Since we want to find the temperature at which the average speed of helium atoms is equal to the escape velocity of Earth (11.2 km/s or 11,200 m/s), we can rearrange the equation as follows:
T = (π * m * v_avg^2) / (8 * k)
Substituting the given values and solving for T:
T = (π * (6.64 x 10^-27 kg) * (11,200 m/s)^2) / (8 * 1.38 x 10^-23 J/K)
T ≈ 21,300 K
Therefore, the temperature at which helium atoms would be able to escape the Earth's gravitational pull is approximately 21,300 Kelvin.
To determine the temperature at which helium atoms would be able to escape Earth's gravitational pull, we need to calculate the average speed of helium atoms and then find the corresponding temperature using the kinetic theory of gases.
1. Calculate the average speed of helium atoms:
The average speed of gas molecules can be calculated using the root mean square (RMS) speed formula:
v_avg = √((3kT) / m)
where:
v_avg = average speed
k = Boltzmann constant (1.38 x 10^-23 J/K)
T = temperature in Kelvin
m = mass of the helium atom (4 atomic mass units)
Considering the average speed required for escape velocity is 11.2 km/s, we convert it to meters per second: 11.2 km/s = 11,200 m/s.
2. Rearrange the formula to solve for temperature (T):
T = (m * v_avg^2)/(3k)
Substituting the values:
T = (4 * (11,200)^2) / (3 * 1.38 x 10^-23)
T ≈ 11,507,246,376,811,594,202 K
Remember to convert the temperature to Celsius or any other scale for practical purposes.
Please note that at such extremely high temperatures, other physical phenomena may occur, such as ionization, rather than just helium atoms escaping due to their average speed.