Question

One mole of $CO_{2}$ at 300 K and 1 atm pressure is heated in a closed vessel so that temperature is 500 K and pressure is 5 atm. Then, it is cooled back to temperature of 300 K and pressure 1 atm. What is the change in internal energy of the gas?

Answer 1

"According to "first law of thermodynamics" change in internal energy (u) = heat liberated (q) + work done (w). The vessel is a closed one. Hence, the "volume remains" the "same".

Hence, work done (w) = 0

Hence, change internal energy (u) = heat liberated (q)

Heat liberated, $q\quad =\quad n\quad \times \quad { C }_{ v }\quad \times \quad T$

Here n is the amount of gas in mole, ${ C }_{ v }$ the "molar heat capacity" at "constant volume".

T represents the temperature change (500 - 200 = 300 K)

At 300 K, the ${ C }_{ v }$ of carbon dioxide is 0.657 kJ/kg.K

Substituting these values in $q\quad =\quad n\quad \times \quad { C }_{ v }\quad \times \quad T$, we have  

$q\quad =\quad 1\quad \times \quad 0.657\quad \times \quad 200\quad =\quad 131.4\quad \sfrac { KJ }.{ kg }$"