Question
A monatomic gas consists of atoms with two internal energy levels: a ground state of degeneracy $g_1$ and a low-lying excited state of degeneracy $g_2$ at an energy $E$ above the ground state. Find the specific heat of this gas.
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We have a monatomic gas with two internal energy levels: a ground state with degeneracy \( g_1 \) and an excited state with degeneracy \( g_2 \) at an energy \( E \) above the ground state. Show more…
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Consider a dilute gas of $N$ noninteracting atoms, in which the atoms have a nondegenerate ground state with energy $E_{0}=0$ and a first excited level that has an energy $E_{1}=\varepsilon$ and degeneracy $g .$ Assume that the second excited state has an energy very high compared with the thermal energy; that is, assume $k T \ll E_{2}$, so that the level $E_{2}$ and all higher levels can be ignored. (a) At temperature $T$, what is the ratio of the number of atoms in the first excited level (degeneracy $g$ ) to the number in the ground state? (b) What is the average energy of an atom in this gas? (c) What is the total energy of the gas? (d) What is the specific heat of the gas?
Statistical Mechanics
Heat Capacities
A monatomic gas is confined to move in two dimensions so that the energy of an atom is $E_{k}=\frac{1}{2} m v_{x}^{2}+\frac{1}{2} m v_{y}^{2}$. What are $C_{V}, C_{p},$ and $\gamma$ for this gas? $\left(C_{p},\right.$ the heat capacity at constant pressure, is equal to $C_{V}+n R$ and $\gamma=C_{P} / C_{V}$ ).
Consider a system made up of $N$ particles. The average energy per particle is given by $\langle E\rangle=\left(\sum E_{i} e^{-E_{i} / k_{B} T}\right) / Z$ where $Z$ is the partition function defined in equation $36.29 .$ If this is a two-state system with $E_{1}=0$ and $E_{2}=E$ and $g_{1}=$ $g_{2}=1,$ calculate the heat capacity of the system, defined as $N(d\langle E\rangle / d T)$ and approximate its behavior at very high and very low temperatures (that is, $k_{\mathrm{B}} T \gg 1$ and $k_{\mathrm{B}} T \ll 1$ ).
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