A piston-cylinder system contains 0.5 moles of an ideal gas initially at a pressure of

A piston-cylinder system contains 0.5 moles of an ideal gas initially at a pressure of $2 \mathrm{~atm}$ and a volume of $10 \mathrm{~L}$ .The gas undergoes an isothermal expansion at a constant temperature of $300 \mathrm{~K}$ until its volume doubles. Calculate the work done by the gas during the expansion and the amount of heat transferred.

Given:
          $\text { Initial pressure }\left(P_{i}\right) =2 \mathrm{~atm}$
          $\text { Initial volume }\left(V_{i}\right) =10 \mathrm{~L}$
          $\text { Final volume }\left(V_{f}\right) =20 \mathrm{~L}$
         $\text { Number of moles }(n) =0.5 \mathrm{moles}$
         $\text { Gas constant }(R) =8.314 \mathrm{~J} /(\mathrm{mol} \mathrm{K})$
           $\text { Temperature }(T) =300 \mathrm{~K}$

Option: 1
-20 L atm

Option: 2
-25 L atm

Option: 3
-24 L atm

Option: 4
-18 L atm

Answers (1)

Step 1: Calculate the change in internal energy using the ideal gas law:

$\mathrm{U=n C_{v} T}$
Where:
         $\mathrm{n }$ is the number of moles
         $\mathrm{C_{v} }$ is the molar heat capacity at constant volume
          $\mathrm{T }$ is the change in temperature

Since the process is isothermal, $\mathrm{T=0 }$ .
$\mathrm{U=0 \mathrm{~J} }$

Step 2: Calculate the work done by the gas using the formula for expansion work:

$\mathrm{W=-U=-P V }$
Where:
$\mathrm{p}$ is the pressure
V is the change in volume
$\mathrm{W=-\left(P_{i}\right)\left(V_{f}-V_{i}\right)}$
Substitute the given values:

$\mathrm{W=-(2 \operatorname{atm})(20 \mathrm{~L}-10 \mathrm{~L})}$
$\mathrm{W=-20 \mathrm{~L} \text { atm }}$

Step 3: Calculate the amount of heat transferred using the first law of thermodynamics:
$\begin{aligned} & U=Q-W \\ & Q=U+W \end{aligned}$
Substitute the calculated value of U and W :
$Q=0 \mathrm{~J}+(-20 \mathrm{~L} \text { atm })$
$Q=-20 \mathrm{~L} \text { atm }$
Answer: During the isothermal expansion, the gas does $-20 \mathrm{~L}$ atm of work, and $-20 \mathrm{~L}$ atm of heat is transferred. Therefore, the correct option is A.