Math Problem Statement

A Carnot engine is operating at temperature of 600K and 900K.

(a) If 16000J of heat energy is absorbed by the engine, how much heat energy is discarded into the cold reservoir?

(b)How much mechanical work is performed by this engine?

(c) Calculate the efficiency of the Carnot engine.

Solution

The Carnot engine's performance can be calculated based on thermodynamic efficiency and energy transfer between the hot and cold reservoirs. Let's solve each part step by step.

Given:

  • Temperature of the hot reservoir (ThT_h) = 900 K
  • Temperature of the cold reservoir (TcT_c) = 600 K
  • Heat absorbed (QhQ_h) = 16000 J

(a) Heat Energy Discarded to Cold Reservoir (QcQ_c)

The efficiency of a Carnot engine is given by:

η=1TcTh\eta = 1 - \frac{T_c}{T_h}

Rearranging for QcQ_c using the relation η=WQh=1QcQh\eta = \frac{W}{Q_h} = 1 - \frac{Q_c}{Q_h}:

Qc=Qh×TcThQ_c = Q_h \times \frac{T_c}{T_h}

Substitute the values:

Qc=16000×600900=16000×23=10666.67JQ_c = 16000 \times \frac{600}{900} = 16000 \times \frac{2}{3} = 10666.67 \, \text{J}

So, the heat energy discarded to the cold reservoir is approximately:

Qc10667JQ_c \approx 10667 \, \text{J}


(b) Mechanical Work Performed by the Engine (WW)

The work WW done by the engine is the difference between the heat absorbed and the heat discarded:

W=QhQcW = Q_h - Q_c

Using the values we found:

W=1600010667=5333JW = 16000 - 10667 = 5333 \, \text{J}

So, the mechanical work performed by the engine is approximately:

W5333JW \approx 5333 \, \text{J}


(c) Efficiency of the Carnot Engine (η\eta)

We already have the efficiency formula for a Carnot engine:

η=1TcTh\eta = 1 - \frac{T_c}{T_h}

Substituting the values:

η=1600900=123=130.333\eta = 1 - \frac{600}{900} = 1 - \frac{2}{3} = \frac{1}{3} \approx 0.333

Expressing this as a percentage:

η33.3%\eta \approx 33.3\%


Summary of Results:

  • (a) Heat energy discarded to cold reservoir, QcQ_c: 10667 J
  • (b) Mechanical work performed, WW: 5333 J
  • (c) Efficiency of the Carnot engine, η\eta: 33.3%

Would you like a detailed explanation for any part?


Related Questions:

  1. How does the efficiency of a Carnot engine change if the temperature of the cold reservoir increases?
  2. If the temperature difference between the hot and cold reservoirs is doubled, how does it affect the work output?
  3. What would be the efficiency of this engine if it operated between 500 K and 800 K instead?
  4. How is the Carnot cycle different from a real-life thermodynamic cycle?
  5. What factors limit the practical efficiency of a Carnot engine?

Tip:

To improve efficiency in any heat engine, increasing the temperature difference between the hot and cold reservoirs generally helps.

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Math Problem Analysis

Mathematical Concepts

Thermodynamics
Carnot Cycle
Efficiency Calculations

Formulas

Carnot efficiency formula: η = 1 - Tc/Th
Heat discarded to cold reservoir: Qc = Qh * (Tc/Th)
Mechanical work: W = Qh - Qc

Theorems

Carnot's theorem

Suitable Grade Level

Undergraduate (Physics or Engineering)