The escape velocity, the velocity required by an object to escape from the
gravitational field is given
V^e = (2gr)^1/2
where r is the radius for Earth (6.37 x 10^6m).
At what temperature will the c of an H2 molecule and of an N2 molecule attain
the escape velocity?
To find the temperature at which the velocity of an H2 molecule and an N2 molecule will attain the escape velocity, we can use the equation for average kinetic energy:
KE = (3/2)kT,
where KE is the kinetic energy, k is the Boltzmann constant (1.38 x 10^-23 J/K), and T is the temperature in Kelvin.
We can set the average kinetic energy equal to the escape velocity to find the temperature at which this occurs.
For an H2 molecule:
V^e = (2gR)^1/2,
KE = (3/2)kT.
Setting the two equations equal to each other:
(2gR)^1/2 = (3/2)kT.
Simplifying the equation:
(2gR)^1/2 = (3/2)(1.38 x 10^-23)T.
Squaring both sides:
2gR = (9/4)(1.38 x 10^-23)^2 T^2.
Simplifying further:
T^2 = (2gR)(4/9)(1.38 x 10^-23)^2.
T = sqrt[(2gR)(4/9)(1.38 x 10^-23)^2].
We can plug in the values for g (acceleration due to gravity) and R (radius of Earth), and calculate the temperature:
g = 9.81 m/s² (acceleration due to gravity on Earth)
R = 6.37 x 10^6 m (radius of Earth)
T = sqrt[(2 * 9.81 * 6.37 x 10^6)(4/9)(1.38 x 10^-23)^2].
Now we can calculate the temperature T using the given values:
T = sqrt[(2 * 9.81 * 6.37 x 10^6)(4/9)(1.38 x 10^-23)^2].
Calculating this equation will give us the temperature at which an H2 molecule will attain the escape velocity. Similarly, we can do the same calculation for an N2 molecule, using its molar mass.
Please note that the molar mass of H2 is 2 g/mol and the molar mass of N2 is 28 g/mol.
To determine the temperature at which the speed of an H2 molecule and an N2 molecule will attain escape velocity, we need to use the concept of kinetic theory of gases.
According to the kinetic theory of gases, the average kinetic energy of a gas molecule is directly proportional to its temperature. The kinetic energy of a gas molecule can be expressed as:
KE = (1/2) * m * v^2
Where KE is the kinetic energy, m is the mass of the molecule, and v is its velocity.
We also know that the escape velocity is given by:
Ve = sqrt(2 * g * r)
Where Ve is the escape velocity, g is the acceleration due to gravity, and r is the radius of the Earth.
For both H2 and N2 molecules to attain the escape velocity, their kinetic energy should be equal to or greater than the energy required to escape. Therefore, we can equate the kinetic energy expression with the escape velocity equation:
(1/2) * m * v^2 = 2 * g * r
Simplifying the equation, we get:
v = sqrt((4 * g * r) / m)
Now, we can determine the temperature at which the H2 and N2 molecules will attain the escape velocity by using the expression for the root mean square velocity (v_rms) of a gas molecule:
v_rms = sqrt((3 * k * T) / m)
Where k is the Boltzmann constant and T is the temperature.
To find the temperature at which the molecules attain escape velocity, we need to equate v_rms with v:
sqrt((3 * k * T) / m) = sqrt((4 * g * r) / m)
Simplifying the equation, we have:
(3 * k * T) / m = (4 * g * r) / m
Dividing both sides by (3 * k / m):
T = (4 * g * r) / (3 * k)
Now we can substitute the values of g (acceleration due to gravity), r (radius of the Earth), and k (Boltzmann constant) to calculate the temperature.
Using the following values:
g = 9.8 m/s^2 (approximate value for acceleration due to gravity)
r = 6.37 x 10^6 m (radius of the Earth)
k = 1.38 x 10^-23 J/K (Boltzmann constant)
T = (4 * 9.8 * 6.37 x 10^6) / (3 * 1.38 x 10^-23)
Now, substitute the values into the equation and calculate the temperature.