Please help. I cant figure out the correct answer! Sparks in air occur when ions
ID: 2000467 • Letter: P
Question
Please help. I cant figure out the correct answer!
Sparks in air occur when ions in the air are accelerated to such a high speed by an electric field that when the ions impact on neutral gas molecules, the neutral molecules become ions. If the electric field strength is large enough, the ionized collision products are themselves accelerated and produce more ions on impact, and so forth. This avalanche of ions is what we call a spark.
nv is the number density of air molecules and d is the molecular diameter.)
E = __________ N/C
Explanation / Answer
Hi,
For this problem I am going to make various suppositions:
1) The air can be worked as an ideal gas (which in the conditions given is something rational).
2) The charge of the ion is similar in magnitude as the one of a proton or electron.
3) The ion is starting its movement from the rest.
4) The molecules of air are like spheres.
With these suppositions we can solve the problem. Of all of them, the second one is the most problematic as the charge of the ion can be different.
Well, the firts thing to do is to define the electric field: E = F/q ; where q is the charge of the ion and F is the electrical force.
So to find E we must find F as we are assuming that q = 1.602*10-19 C.
In order to find F we can remember Newton's Second Law, which says that : F = ma ; where m is the mass of the ion and a is the acceleration of it.
So now we must find a, to do that we can remember some equations from kinetics. For instance the one with velocity as a function of acceleration: v2 = vo2 + 2a(x-x0) ; where v is the velocity, vo is the initial velocity, x is the final position and x0is the initial position.
Up until now we must find v and x to solve the problem. The initial values are not a problem because as we assume the ion is starting ist movement from the rest, the initial velocity is cero. Besides, we can choose the initial position as cero as well.
To find v we should remember the definition of kinetic energy, which is: K = (1/2)*m*v2 ; as K is given we can express the velocity in terms of the kinetic energy: v = (2*K*m)1/2
To find x we rely on the formula given for the mean free path (as x is the distance traveled by the ion before hitting a molecule). In that formula we have two things we are lacking: d and nv
In the case of d we can assume that, as the air molecules are like spheres, its cross-sectional area can be calculated through: A = 4**r2 ; and if d = 2r then we have the following: d = (A/)1/2 ; so d is equal to:
d = (0.092*10-9 m2 /)1/2 = 5.412*10-6 m
In the case of nv we can use the equation for ideal gas, if we define nv as: nv = n/V ; where n is the number of mols and V is the volumen measured in m3.
The equation for the ideal gas is: PV = nTR; and therefore: nv = P/(RT) so this parameter is equal to:
nv = (101325 Pa)/(8.314 Pa*m3/mol*K * 300K) = 40.62 mol/m3 ; however, as we want to calculate the density of molecules we should have something like molecules/m3 as units. To do this we can multiply our result for the Avogadro constant (NA)
Now we can calculate x as follow:
x =1/(21/2*nv*NA**d2) = 1/(1.414*40.62*(6.02*1023)**(5.412*10-6)2) = 3.14*10-16 m
Now we replace the definition of velocity in terms of kinetic energy in the equation of acceleration and we expressed the acceleration in terms of the other parameters, if we do that we get the following.
a = K/(mx)
After that we can replace this definition of acceleration in the equation with the force, and by doing that we get:
F= K/m
Finally, we put this definition of force in the equation with electrical field and we solve the problem. Of course, as it was said before the magnitude of E depends on the magnitude of q.
E = K/(x*q) = (0.94)*1.602*10-19 J / ((3.14*10-16 m)*(1.602*10-19 C)) = 3*1015 N/C.
A way to express the electric field in terms of any charge would be:
E = (4.80*10-4)/ Q [N/C]
I hope it helps, and sorry for any mistake I could have committed.