Question

Two moles of an ideal gas $\left( {{{\text{C}}_{\text{V}}} = 5/2{\text{R}}} \right)$ was compressed adiabatically against constant pressure of 2 atm which was initially at 350 K and 1 atm pressure. The work involve in the process is equal to
A.250 R
B.300 R
C.400 R
D.500 R

Answer 1

For this question we must know the relation between work and temperature for adiabatic processes. Using the given set of systems we will first calculate the final temperature and then using the same formula we will calculate work done in terms of R.
Formula used: ${\text{w}} = {\text{n}}{{\text{C}}_{\text{V}}}\left( {{{\text{T}}_2} - {{\text{T}}_1}} \right) = - {{\text{P}}_{{\text{ext}}}} \times {\text{nR}}\left[ {\dfrac{{{{\text{T}}_2}}}{{{{\text{P}}_2}}} - \dfrac{{{{\text{T}}_1}}}{{{{\text{P}}_1}}}} \right]$
Here w is the work done, n is the number of moles, ${{\text{C}}_{\text{V}}}$ is heat capacity at constant volume, R is the universal gas constant, P represents pressure and T represents temperature.

Complete step by step answer:
Using the given values we will first calculate the final temperature of the system because that will give us the value of work.
${{\text{P}}_{{\text{ext}}}} = {{\text{P}}_2} = 2{\text{ atm}}$ because external pressure is same as final temperature.
${\text{2}} \times \dfrac{5}{2}{\text{R}}\left( {{{\text{T}}_2} - 350} \right) = - 2 \times 2{\text{R}}\left[ {\dfrac{{{{\text{T}}_2}}}{2} - \dfrac{{350}}{1}} \right]$
Solving the above equation and rearranging we will get:
${\text{5}}{{\text{T}}_2} - 1750 = - 2{{\text{T}}_2} + 1400$
On simplifying the above equation we will get:
${{\text{T}}_2} = 450$
Now we will use this temperature into the formula of work as:
${\text{w}} = 2 \times \dfrac{5}{2}{\text{R}}\left( {{\text{450}} - 350} \right) = 500{\text{R}}$

The correct option is D.

Note:
Adiabatic processes that in which the heat change is 0. That is no heat enters a system or no heat leaves the system. An ideal gas is a gas that follows all the gas laws under all conditions of temperature and pressure. In actual practice no gas is ideal gas, it is only a theoretical or hypothetical concept. When the work is done upon the system when the system compresses but if the work is done by the system the system expands. R is the universal gas constant which is the ratio of heat capacity at constant pressure to the heat capacity at constant volume.