Question

A monatomic gas at pressure ${{P}_{1}}$​ and volume ${{V}_{1}}$​ is compressed adiabatically to $\dfrac{1}{8}th$ of its original volume. What is the final pressure of the gas?
$\begin{align}
  & (A)64{{P}_{1}} \\
 & (B){{P}_{1}} \\
 & (C)16{{P}_{1}} \\
 & (D)32{{P}_{1}} \\
\end{align}$

Answer 1

Hint: An ideal process is the process in which there is no transfer of heat or mass between a thermodynamic system and its surrounding. The adiabatic compression of a gas causes an increase in temperature of the gas. Adiabatic expansion against pressure, or a spring, causes a drop by temperature. In contrast, free expansion is an isothermal process for a perfect gas.

Formula used:
For adiabatic process:
\[P{{V}^{\gamma }}=~constant\]
Where:
$P$= Pressure of the gas
$V$= Velocity of the gas
$\gamma $= Gas constant

Complete step-by-step answer:
As we know for Ideal gas equation, an adiabatic process is
\[P{{V}^{\gamma }}=~constant\text{ }~\text{ }~\]
or
\[{{P}_{1}}{{V}_{2}}^{\gamma }={{P}_{2}}{{V}_{2}}^{\gamma }\] ​

For monoatomic gas ​
\[\gamma =\dfrac{5}{3}\]
\[\begin{align}
  & {{P}_{1}}{{V}_{1}}^{\dfrac{5}{3}}={{P}_{2}}{{\left( \dfrac{{{V}_{1}}}{8} \right)}^{\dfrac{5}{3}}} \\
 & \Rightarrow {{P}_{2}}={{P}_{1}}\times {{\left( 2 \right)}^{5}}=32{{P}_{1}} \\
\end{align}\]​

So, the correct answer is “Option D”.

Additional Information: Monatomic gas is composition of molecules that includes single atoms, like helium or sodium vapour, and during this way different from diatomic, triatomic, or, generally, polyatomic gases. The thermodynamic behaviour of a monatomic gas within the ordinary temperature range is very simple because it's free from the rotational and vibrational energy components characteristic of polyatomic gases; thus, its heat capacity is independent of temperature and molecular (here, atomic) weight, and its entropy (a measure of disorder) depends only on temperature and relative molecular mass. For a perfect gas at constant pressure, it takes more heat to realize an equivalent natural process than it does at constant volume. At constant volume all the warmth added goes into raising the temperature. At constant pressure a number of the warmth goes to doing work.

Note: It is usually applied to gases: a monatomic gas is one during which atoms aren't sure to one another. The thermodynamic behaviour of a monatomic gas is very simple in comparison to polyatomic gases because it's freed from any rotational or vibrational energy.