to start a nuclear fusion reaction, two hydrogen atoms of charge 1.602x10^-19 and mass 1.67x10^-27kg must be fired at each other. If each particle has an initial velocity of 2.7x10^6m/s when released, what is their minimum seperation?
KE =PE
mv²/2= ke²/r ,
r= 2•k•e²/m•v²,
where
k =9•10^9 N•m²/C²,
e =1.6•10^-19 C,
m =1.66•10^-27 kg.
To find the minimum separation between the two hydrogen atoms when they collide to start a nuclear fusion reaction, we can make use of the conservation of kinetic energy.
1. First, let's determine the initial kinetic energy of each hydrogen atom. The kinetic energy (KE) can be calculated using the formula:
KE = (1/2)mv^2
Where m is the mass of the hydrogen atom and v is its velocity.
Substituting the given values:
m = 1.67x10^-27 kg
v = 2.7x10^6 m/s
KE = (1/2)(1.67x10^-27 kg)(2.7x10^6 m/s)^2
2. Calculate the initial total kinetic energy of the two hydrogen atoms by multiplying the individual kinetic energies by two since there are two atoms:
Total KE = 2 * KE
3. The initial total kinetic energy is equal to the electrical potential energy when the two hydrogen atoms are at their minimum separation (r). The electrical potential energy (PE) is given as:
PE = k * (Q^2/r)
Where k is the electrostatic constant, Q is the charge of the hydrogen atom, and r is the separation distance between them.
4. Rearranging the equation, we can find the minimum separation (r):
r = k * (Q^2 / Total KE)
k = 8.988x10^9 N m^2/C^2 (Coulomb's constant)
Q = 1.602x10^-19 C
Substituting these values, we can calculate the minimum separation (r).
I will calculate the final answer.
To determine the minimum separation between the two hydrogen atoms in order to initiate a nuclear fusion reaction, we can use the principles of conservation of energy and momentum.
First, we need to calculate the kinetic energy of each hydrogen atom. The kinetic energy (KE) is given by the formula KE = 1/2 * mass * velocity^2.
Given:
Charge of hydrogen atom = 1.602x10^-19 C
Mass of hydrogen atom = 1.67x10^-27 kg
Initial velocity of each hydrogen atom = 2.7x10^6 m/s
Calculating the kinetic energy for each hydrogen atom:
KE = 1/2 * (1.67x10^-27 kg) * (2.7x10^6 m/s)^2
Next, we can calculate the potential energy (PE) of the interaction between the two hydrogen atoms. The potential energy of the interaction is given by the formula PE = (k * (charge1 * charge2)) / distance.
Given:
Electric charge of each hydrogen atom = 1.602x10^-19 C
Distance between the hydrogen atoms = x (to be determined)
For hydrogen atoms, k = 8.99x10^9 Nm^2/C^2 (Coulomb's constant).
Now, let's equate the kinetic energy to the potential energy by assuming that the kinetic energy is fully converted into potential energy at the minimum separation distance.
1/2 * (1.67x10^-27 kg) * (2.7x10^6 m/s)^2 = (8.99x10^9 Nm^2/C^2 * (1.602x10^-19 C)^2) / x
Simplifying the equation, we can solve for the value of x (the minimum separation distance) using basic algebraic steps.
x = (8.99x10^9 Nm^2/C^2 * (1.602x10^-19 C)^2) / (1/2 * (1.67x10^-27 kg) * (2.7x10^6 m/s)^2)
Evaluating the above expression should give you the minimum separation distance between the two hydrogen atoms.