Question

A gas is allowed to expand in a well insulated container against constant external pressure of $2.5\mathrm{atm}$ from an initial volume of $2.50\mathrm{L}$ to a final volume of $4.50\mathrm{L}$. The change in internal energy $\mathrm{\Delta }\mathrm{U}$ of the gas in joules will be ?

• A. $+505\mathrm{J}$
• B. $+1136.25\mathrm{J}$
• C. $-500\mathrm{J}$
• D. $-505\mathrm{J}$

Answer 1

The correct option is D

$-505\mathrm{J}$

Explanation for the correct option:

Option(D):$-505\mathrm{J}$

Step 1: Analyzing Given data:

The constant external pressure is given as= $2.5\mathrm{atm}$

The initial volume =$2.50\mathrm{L}$

and final volume = $4.50\mathrm{L}$

Step 2: Formula for calculating $\mathrm{\Delta U}$:

According to the first law of thermodynamics, the formula related to the change in the internal energy can be given as:

$\mathrm{\Delta U}=\mathrm{q}+\mathrm{W}$

(where$\mathrm{\Delta U}$ = change in internal energy of the system

$\mathrm{q}=$ heat transfer between the system

$\mathrm{W}=$work done by the system)

Step 3: Calculation of $\mathrm{\Delta U}$:

Also, as per the given question, the gas is allowed to expand in a well-insulated container, hence, heat transfer i.e $\mathrm{q}=0$

Hence,

$$\begin{gathered} \mathrm{\Delta U}=\mathrm{W}=-\mathrm{P\Delta V} \\ \Rightarrow \mathrm{\Delta U}=\mathrm{W}=-2.5\mathrm{atm}\mathrm{x}(4.50\mathrm{L}–2.50\mathrm{L}) \\ \Rightarrow \mathrm{\Delta U}=\mathrm{W}=-5.0\mathrm{L}\mathrm{atm} \\ \Rightarrow \mathrm{\Delta U}=\mathrm{W}=-5.0\mathrm{L}\mathrm{atm}\mathrm{x}(101.33\mathrm{J}/1\mathrm{L}\mathrm{atm}) \\ \Rightarrow \mathrm{\Delta U}=-505\mathrm{J} \end{gathered}$$

Hence, the change in internal energy $\mathrm{\Delta }\mathrm{U}$ of the gas in joules will be $-505\mathrm{J}$.

Thus, option D is correct.