Question

A mass of diatomic gas ($\gamma = 1.4$) at a pressure of 2 atm is compressed adiabatically so that its temperature rises from $27^\circ C$ to $927^\circ C$. The pressure of the gas in the final state is
A.) 28 atm
B.) 68.7 atm
C.) 256 atm
D.) 8 atm

Answer 1

Hint: The equation for an adiabatic process describes the relation between pressure and volume which can be converted in terms of temperature and pressure using the ideal gas equation. Relating the initial and final values gives the required value of pressure.

Formula Used:
For an adiabatic process, temperature and pressure are related to each other by the following relation:
$\dfrac{{{T^\gamma }}}{{{P^{\gamma - 1}}}} = $ constant …(i)
where $\gamma $ is called the adiabatic index which is the ratio of specific heat of the gas measured at constant pressure to the specific heat of gas measured at constant volume.

Detailed step by step solution:
In thermodynamics, an adiabatic process is that process in which there is no transfer of heat or mass between the system and the surroundings.

The equation used to describe an adiabatic process is given as

$P{V^\gamma } = $ constant …(ii)

We are given a diatomic gas whose adiabatic index is

$\gamma = 1.4$

Its initial temperature and pressure are given as

$
  {P_1} = 2atm \\
  {T_1} = 27^\circ C = 27 + 273 = 300K \\
$

This gas is made to undergo adiabatic compression due to which its temperature increases to the following final value:

${T_2} = 927^\circ C = 927 + 273 = 1200K$

We need to find the final pressure at this temperature which can be found using the relation between temperature and pressure for adiabatic processes. By using ideal gas equation, $PV = RT$ in equation (ii) and substituting the value of volume, we obtain equation (i)

$P{\left( {\dfrac{T}{P}} \right)^\gamma } = $constant
$ \Rightarrow \dfrac{{{T^\gamma }}}{{{P^{\gamma - 1}}}} = $constant
For initial and final conditions, we can write this equation as

$\dfrac{{{T_1}^\gamma }}{{{P_1}^{\gamma - 1}}} = \dfrac{{{T_2}^\gamma }}{{{P_2}^{\gamma - 1}}}$

Substituting all the known values in this equation, we get

$
  \dfrac{{{{\left( {300} \right)}^{1.4}}}}{{{{\left( 2 \right)}^{1.4 - 1}}}} = \dfrac{{{{\left( {1200} \right)}^\gamma }}}{{{P_2}^{\gamma - 1}}} \\
   \Rightarrow {\left( {\dfrac{{{P_2}}}{2}} \right)^{0.4}} = {\left( {\dfrac{{1200}}{{300}}} \right)^{1.4}} \\
   \Rightarrow \left( {\dfrac{{{P_2}}}{2}} \right) = {\left( {\dfrac{{1200}}{{300}}} \right)^{\dfrac{{1.4}}{{0.4}}}} \\
   \Rightarrow {P_2} = 2 \times {\left( 4 \right)^{3.5}} \\
   \Rightarrow {P_2} = 256atm \\
$

Hence the correct answer is option C.

Note:
1. In order to ensure that an adiabatic process takes place, the thermodynamic system must be perfectly insulated from its surroundings.
2. When a gas compressed adiabatically, the internal energy of the gas increases which leads to an increase in temperature of gas.