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# A 2.00-mol sample of a diatomic ideal gas expands slowly and adiabatically from a pressure of $5.00 \mathrm{~atm}$ and a volume of $12.0 \mathrm{~L}$ to a final volume of $30.0 \mathrm{~L}$. What are the initial and final temperatures?

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Concepts and Principles

• When an ideal gas undergoes a slow adiabatic volume change ($\textcolor{#c34632}{Q=0}$), its pressure and volume are related by:

$\begin{equation*} PV^{\gamma}=\text{constant}\tag{1} \end{equation*}$

Where $\textcolor{#c34632}{\gamma=C_P/C_V}$ is the ratio of molar specific heats for the gas. For a diatomic gas, $\textcolor{#c34632}{\gamma=1.4}$.

• Equation of State of an Ideal Gas: an ideal gas is one for which the pressure $\textcolor{#c34632}{P}$, volume $\textcolor{#c34632}{V}$, and temperature $\textcolor{#c34632}{T}$ are related by:

$\begin{equation*} PV=nRT\tag{1} \end{equation*}$

Where,

$\textcolor{#c34632}{n}$ is the number of moles of the gas present.
$\textcolor{#c34632}{R}$ is the universal gas constant and has a value $\textcolor{#c34632}{8.314\;\mathrm{J/mol\cdot K}}$.

• The change in the internal energy of $\textcolor{#c34632}{n}$ moles of a confined ideal gas that undergoes a temperature change $\textcolor{#c34632}{\Delta T}$ due to any process is:

$\begin{equation*} \Delta E_{\text{int}}=nC_V\Delta T\tag{3} \end{equation*}$

Where $\textcolor{#c34632}{C_V}$ is the molar specific heat at constant volume for the gas.

• First Law of Thermodynamics: it is the principle of conservation of energy for a thermodynamic process which is expressed as:

$\begin{equation*} \Delta E_{\text{int}}=Q+W\tag{4} \end{equation*}$

Where $\textcolor{#c34632}{\Delta E_{\text{int}}}$ is the internal energy of the system, $\textcolor{#c34632}{Q}$ is the energy transferred into the system by heat between the system and its surroundings, and $\textcolor{#c34632}{W}$ is the work done on the system.

In this problem, I am displaying rounded intermediate values for practical purposes. However, calculations are made using the unrounded values.

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