A 3-dimensional ideal gas obeys pV = 3/2E, where p is the pressure, V is the volume, and E is the internal energy

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1. We saw in class that a 3-dimensional ideal gas obeys pV = 3/2E, where p is the pressure, V is the volume, and E is the internal energy (average kinetic energy). Derive the corresponding formula for a two-dimensional ideal gas, i.e. a collection of noninteracting particles that moves on a plane. (Hint: suppose a two-dimensional ideal gas of N particles, with internal energy E, is confined by one-dimensional "walls" to an area A; the pressure is defined as the force per unit length).
2. Each r seconds a particle, which was initially x = 0, jumps either left or right (each with probability 1/2) a distance a. At time tn= nr, the particle is at location xk = ka with probability P(n, k). Calculate P(n, k), and <(k)^2>.