The average kinetic energy of a particle in a cubical box of side l, whose top and bottom sides are parallel to the earth's surface, is kB * t + (1/3) * mg l.
The average kinetic energy of a particle is given by:
KE = (1/2) * mv^2
where m is the mass of the particle and v is its velocity.
The velocity of a particle can be found using the following equation:
v = sqrt(2KE/m)
The total kinetic energy of the gas is given by:
KE = N * (1/2) * mv^2
We can now substitute the second equation into the first equation to get:
KE = N * (1/2) * m * (sqrt(2KE/m))^2
This simplifies to:
KE = (3/2) * N * KE
Solving for KE, we get:
KE = N * KE / (3/2)
The average kinetic energy of a particle is therefore:
KE = (1/2) * mv^2 = (3/2) * KE / N
In this case, the particles are monatomic, so m is the mass of a single atom.
The gas is in thermal equilibrium, so the average kinetic energy of a particle is related to the temperature of the gas by the following equation:
KE = (3/2) * kB * t
where kB is the Boltzmann constant.
The acceleration due to gravity is g, so the gravitational potential energy of a particle is:
PE = mgz
where z is the height of the particle above the earth's surface.
The total gravitational potential energy of the gas is given by:
PE = N * mgz
The average gravitational potential energy of a particle is therefore:
PE = N * mgz / N = mgz
The total energy of a particle is given by:
E = KE + PE
Substituting the previous equations into this equation, we get:
E = (3/2) * kB * t + mgz
The gas is in a cubical box of side l, so the height of a particle above the earth's surface is z = l/2. Substituting this into the previous equation, we get:
E = (3/2) * kB * t + (1/2) * mg l
The total energy of the gas is given by:
E = N * ((3/2) * kB * t + (1/2) * mg l)
The average energy of a particle is therefore:
E = N * ((3/2) * kB * t + (1/2) * mg l) / N = (3/2) * kB * t + (1/2) * mg l
Substituting the previous equation into the equation for the average kinetic energy of a particle, we get:
KE = (1/2) * mv^2 = (3/2) * kB * t + (1/2) * mg l / (3/2) = kB * t + (1/3) * mg l
Therefore, the average kinetic energy of a particle in a cubical box of side l, whose top and bottom sides are parallel to the earth's surface, is kB * t + (1/3) * mg l.