The thermal energy of a monatomic gas is the total kinetic energy of the gas atoms. This kinetic energy can be described as the combination of three independent modes of energy in the system since we can write the kinetic energy of a particle as \( \frac{1}{2} m v^{2}=\frac{1}{2} m v_{x}^{2}+\frac{1}{2} m v_{y}^{2}+\frac{1}{2} m v_{z}^{2} \). The motion of the atoms in the \( \mathrm{x}-\mathrm{y} \)-, and z -directions are the three independent modes of energy stored in this system. The equipartition theorem states that each independent mode of energy storage, or degree of freedom, in a system (temperature \( T \), number of particles \( N \), or number of moles \( n \) ) stores an equal amount of energy, \( \frac{1}{2} N k_{B} T=\frac{1}{2} n R T \) per degree of freedom.
a. The thermal energy of a monatomic gas is \( E_{\mathrm{th}}=\frac{3}{2} n R T \). Since \( \Delta E_{\mathrm{th}}=n C_{\mathrm{V}} \Delta T \), what is \( C_{\mathrm{V}} \) for a monatomic gas?
b. Consider a two-dimensional monatomic gas. This is a system of atoms that can move in the \( x \) and \( y \)-directions, but unable to move in the \( z \)-direction.
i. What is the thermal energy of \( n \) moles of a two-dimensional monatomic gas with temperature \( T \) ?
ii. What is \( C_{V} \) for a two-dimensional monatomic gas?
