Question

A gas expands from a volume of \(1\) dm\(^3\) to \(1.25\) dm\(^3\) under a pressure of \(1\) bar. Find the change in internal energy if \(150\) J of heat is supplied to the system.

• A. \(125\) J
• B. \(175\) J
• C. \(-125\) J
• D. \(-175\) J

Hint:

Work done by the gas is \(P\Delta V\); then use the first law \(\Delta U = q + w\).

Answer 1

The Correct Option is A

Step 1: Plan:
We need $\Delta U$ for a gas that absorbs heat and expands. Energy conservation says: energy gained as heat minus energy spent as expansion work equals the rise in internal energy.

Step 2: Expansion work in joules:
Pressure $= 1$ bar $= 10^5$ Pa. Volume change $= 0.25$ dm$^3$ $= 0.25 \times 10^{-3}$ m$^3$.
\[ W_{\text{by gas}} = P\Delta V = 10^5 \times 0.25\times 10^{-3} = 25\ \text{J} \]

Step 3: Energy balance:
\[ \Delta U = q - W_{\text{by gas}} = 150 - 25 = 125\ \text{J} \]
This is the same as writing $\Delta U = q + w$ with $w = -25$ J.

Step 4: Checking the distractors:
Option B ($175$ J) forgets that the gas loses energy while pushing the surroundings back. Options C and D are negative, but the gas takes in more energy than it gives away, so $\Delta U$ cannot be negative.

Final Answer:
Net energy gained by the gas is $125$ J, so option (A) is correct. \[ \boxed{125\ \text{J}} \]