Question

A gas is allowed to expand in an insulated container against a constant external pressure of 2.5 atm from 2.5 L to 4.5 L, the change in internal energy of the gas in joules is

• A. $-836.3\text{ J}$
• B. $-1136.2\text{ J}$
• C. $-450\text{ J}$
• D. $-506.5\text{ J}$

Hint:

The word "insulated" is the most important clue here! It immediately tells you that $q=0$, turning the first law of thermodynamics into the simple equation $\Delta U = W$.

Answer 1

The Correct Option is D

Step 1: Understanding the Question:
We need to calculate the change in internal energy ($\Delta U$) of a gas undergoing an irreversible expansion in an insulated container.

Step 2: Key Formula or Approach:
According to the first law of thermodynamics: $\Delta U = q + W$.
Since the container is insulated, the process is adiabatic, meaning no heat is exchanged with the surroundings ($q = 0$).
Therefore, $\Delta U = W$.
The work done ($W$) against a constant external pressure is given by: $W = -P_{ext}\Delta V = -P_{ext}(V_2 - V_1)$.
Finally, to convert the work from Liter-atmospheres to Joules, we use the conversion factor: $1\text{ L atm} = 101.325\text{ J}$.

Step 3: Detailed Explanation:
First, calculate the change in volume ($\Delta V$):
$$\Delta V = 4.5\text{ L} - 2.5\text{ L} = 2.0\text{ L}$$
Now, calculate the work done in L atm:
$$W = -2.5\text{ atm} \times 2.0\text{ L}$$
$$W = -5.0\text{ L atm}$$
Since $\Delta U = W$, the change in internal energy is $-5.0\text{ L atm}$.
Convert this value to Joules:
$$\Delta U = -5.0\text{ L atm} \times 101.325\text{ J/L atm}$$
$$\Delta U = -506.625\text{ J}$$
Rounding to the nearest decimal matching the options gives $-506.5\text{ J}$.

Step 4: Final Answer:
The change in internal energy is approximately $-506.5\text{ J}$, which corresponds to option (D).