Question

During one cycle the working fluid in an engine engages in two work interactions: 15 kJ to the fluid and 45 kJ from the fluid, and three heat interactions, two of which are known: 75 kJ to the fluid and 40 kJ from the fluid. The magnitude and direction of the third heat transfer is

• A. 5 kJ from the system
• B. 55 kJ into the system
• C. 5 kJ into the system
• D. -5 kJ from the system

Hint:

In a cyclic process, use the first law (\( Q_{\text{net}} = W_{\text{net}} \)) to find unknown heat or work transfers, ensuring proper signs for direction.

Answer 1

The Correct Option is D

Step 1: Understand the cyclic process and apply the first law.
The process is cyclic, so the change in internal energy \( \Delta U = 0 \). The first law of thermodynamics for a cyclic process is: \[ Q_{\text{net}} = W_{\text{net}}, \] where \( Q_{\text{net}} \) is the net heat transfer, and \( W_{\text{net}} \) is the net work done. Step 2: Calculate the net work transfer.
Work interactions: 15 kJ to the fluid (work done on the system, negative work done by the system): \( W_1 = -15 \, \text{kJ} \), 45 kJ from the fluid (work done by the system): \( W_2 = +45 \, \text{kJ} \). Net work done by the system: \[ W_{\text{net}} = W_2 + W_1 = 45 + (-15) = 30 \, \text{kJ}. \] Step 3: Calculate the known net heat transfer.
Heat interactions:
75 kJ to the fluid (heat added to the system): \( Q_1 = +75 \, \text{kJ} \),
40 kJ from the fluid (heat rejected by the system): \( Q_2 = -40 \, \text{kJ} \),
Third heat transfer \( Q_3 \) (unknown). Net known heat transfer: \[ Q_1 + Q_2 = 75 - 40 = 35 \, \text{kJ}. \] Step 4: Determine the third heat transfer.
For a cyclic process: \[ Q_{\text{net}} = W_{\text{net}}, \] \[ Q_1 + Q_2 + Q_3 = W_{\text{net}}, \] \[ 75 - 40 + Q_3 = 30, \] \[ 35 + Q_3 = 30, \] \[ Q_3 = 30 - 35 = -5 \, \text{kJ}. \] - A negative value means 5 kJ of heat is transferred out of the system (from the system). Step 5: Interpret the magnitude and direction.
Magnitude: 5 kJ,
Direction: From the system (since \( Q_3 \) is negative).
The option "-5 kJ from the system" indicates a heat transfer of 5 kJ out of the system, which matches.
Step 6: Select the correct answer.
The third heat transfer is -5 kJ from the system, matching option (4).