Question

A gas is compressed at a constant pressure of 50 N/m\(^2\) from a volume of 10 m\(^3\) to a volume of 4 m\(^3\). Energy of 100 J is then added to the gas by heating. Its internal energy is

• A. Increases by 400 J
• B. Increases by 200 J
• C. Increases by 100 J
• D. Decreases by 200 J

Hint:

Be very careful with the sign conventions in thermodynamics. \(Q\) is positive when heat is added to the system. \(W\) is positive when work is done by the system. In this case, compression means the system's volume decreases (\(\Delta V \lt 0\)), so the work done by the system is negative.

Answer 1

The Correct Option is A

Step 1: Understanding the Question:
We are analyzing a thermodynamic process where a gas is compressed and then heated. We need to find the total change in its internal energy.

Step 2: Key Formula or Approach:
We will use the First Law of Thermodynamics, which states that the change in internal energy (\(\Delta U\)) of a system is equal to the heat added to the system (\(Q\)) minus the work done by the system (\(W\)).
\[ \Delta U = Q - W \]
The work done by the gas during a constant pressure (isobaric) process is given by:
\[ W = P \Delta V = P(V_{final} - V_{initial}) \]

Step 3: Detailed Explanation:
First, let's identify the given quantities:
- Heat added to the gas, \(Q = +100\) J.
- Constant pressure, \(P = 50\) N/m\(^2\).
- Initial volume, \(V_{initial} = 10\) m\(^3\).
- Final volume, \(V_{final} = 4\) m\(^3\).
Next, calculate the work done

by the gas:
\[ W = P(V_{final} - V_{initial}) = 50 \text{ N/m}^2 \times (4 \text{ m}^3 - 10 \text{ m}^3) \]
\[ W = 50 \times (-6) = -300 \text{ J} \]
The negative sign indicates that work is not done by the gas, but rather work is done

on the gas during compression.
Now, apply the First Law of Thermodynamics to find the change in internal energy:
\[ \Delta U = Q - W \]
\[ \Delta U = 100 \text{ J} - (-300 \text{ J}) \]
\[ \Delta U = 100 + 300 = 400 \text{ J} \]
Since \(\Delta U\) is positive, the internal energy increases.

Step 4: Final Answer:
The internal energy of the gas increases by 400 J.