A Power Plant Burns Coal At 835 K And Dumps Heat Into The Air At 288 K. The Plant Runs At The Carnot Efficiency.How Much Input Heat Is Required For The Plant To Produce 235,000 J Of Work? Q = [ ? ] J Q = [?] \, \text{J} Q = [ ?] J

Mar 12, 2025 by ADMIN 231 views

A Power Plant's Efficiency: Understanding the Carnot Cycle

Introduction

The Carnot cycle is a theoretical thermodynamic cycle that provides an upper limit to the efficiency of any heat engine. It is a fundamental concept in understanding the efficiency of power plants, refrigerators, and other heat transfer systems. In this article, we will explore the Carnot cycle and its application to a power plant that burns coal at 835 K and dumps heat into the air at 288 K.

The Carnot Cycle

The Carnot cycle is a four-stage process that consists of isothermal expansion, adiabatic expansion, isothermal compression, and adiabatic compression. The cycle is represented by a series of thermodynamic processes that involve heat transfer and work done by the system.

Stage 1: Isothermal Expansion

In the first stage, the system is in contact with a heat reservoir at a temperature of $T_H$ . The system expands isothermally, absorbing heat from the reservoir. This process is represented by the equation:

Q_H = T_H \Delta S

where $Q_H$ is the heat absorbed from the reservoir, $T_H$ is the temperature of the reservoir, and $\Delta S$ is the change in entropy.

Stage 2: Adiabatic Expansion

In the second stage, the system expands adiabatically, meaning that there is no heat transfer between the system and its surroundings. During this process, the temperature of the system decreases.

Stage 3: Isothermal Compression

In the third stage, the system is in contact with a heat reservoir at a temperature of $T_C$ . The system compresses isothermally, releasing heat to the reservoir. This process is represented by the equation:

Q_C = T_C \Delta S

where $Q_C$ is the heat released to the reservoir, $T_C$ is the temperature of the reservoir, and $\Delta S$ is the change in entropy.

Stage 4: Adiabatic Compression

In the fourth stage, the system compresses adiabatically, meaning that there is no heat transfer between the system and its surroundings. During this process, the temperature of the system increases.

The Carnot Efficiency

The Carnot efficiency is a measure of the efficiency of a heat engine. It is defined as the ratio of the work done by the engine to the heat absorbed from the reservoir. The Carnot efficiency is represented by the equation:

\eta = 1 - \frac{T_C}{T_H}

where $\eta$ is the Carnot efficiency, $T_C$ is the temperature of the cold reservoir, and $T_H$ is the temperature of the hot reservoir.

Application to a Power Plant

A power plant burns coal at 835 K and dumps heat into the air at 288 K. The plant runs at the Carnot efficiency. We want to find out how much input heat is required for the plant to produce 235,000 J of work.

Given Values

$T_H = 835$ K
$T_C = 288$ K
$W = 235,000$ J

Unknown Value

$Q_H = ?$ J

Solution

We can use the Carnot efficiency equation to find the ratio of the heat absorbed from the reservoir to the work done by the engine:

\frac{Q_H}{W} = \frac{1}{\eta}

Substituting the values of $T_H$ , $T_C$ , and $W$ , we get:

\frac{Q_H}{235,000} = \frac{1}{1 - \frac{288}{835}}

Simplifying the equation, we get:

\frac{Q_H}{…" style="color:#cc0000">\frac{Q_H}{235,000} = \frac{1}{1 - 0.345}$ $\frac{Q_H}{235,000} = \frac{1}{0.655} </span></p> <p class='katex-block'><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><msub><mi>Q</mi><mi>H</mi></msub><mo>=</mo><mn>235</mn><mo separator="true">,</mo><mn>000</mn><mo>×</mo><mn>1.53</mn></mrow><annotation encoding="application/x-tex">Q_H = 235,000 \times 1.53 </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8778em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathnormal">Q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.08125em;">H</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8389em;vertical-align:-0.1944em;"></span><span class="mord">235</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">000</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1.53</span></span></span></span></span></p> <p class='katex-block'><span class="katex-error" title="ParseError: KaTeX parse error: Can&#x27;t use function &#x27;{{content}}amp;#x27; in math mode at position 14: Q_H = 358,650$̲ J Therefore, …" style="color:#cc0000">Q_H = 358,650$ J Therefore, the input heat required for the plant to produce 235,000 J of work is 358,650 J. ## Conclusion In this article, we have explored the Carnot cycle and its application to a power plant that burns coal at 835 K and dumps heat into the air at 288 K. We have used the Carnot efficiency equation to find the ratio of the heat absorbed from the reservoir to the work done by the engine. We have also calculated the input heat required for the plant to produce 235,000 J of work. The result shows that the input heat required is 358,650 J. ## References * Carnot, S. (1824). Reflections on the Motive Power of Fire and on Machines Fitted to Develop That Power. Bachelier. * Feynman, R. P. (1963). The Feynman Lectures on Physics. Addison-Wesley. * Halliday, D., Resnick, R., &amp; Walker, J. (2013). Fundamentals of Physics. John Wiley &amp; Sons.&lt;br/&gt; # A Power Plant&#x27;s Efficiency: Understanding the Carnot Cycle - Q&amp;A ## Introduction In our previous article, we explored the Carnot cycle and its application to a power plant that burns coal at 835 K and dumps heat into the air at 288 K. We calculated the input heat required for the plant to produce 235,000 J of work. In this article, we will answer some frequently asked questions related to the Carnot cycle and its application to power plants. ## Q&amp;A ### Q1: What is the Carnot cycle? A1: The Carnot cycle is a theoretical thermodynamic cycle that provides an upper limit to the efficiency of any heat engine. It is a four-stage process that consists of isothermal expansion, adiabatic expansion, isothermal compression, and adiabatic compression. ### Q2: What are the stages of the Carnot cycle? A2: The stages of the Carnot cycle are: 1. Isothermal expansion: The system expands isothermally, absorbing heat from the reservoir. 2. Adiabatic expansion: The system expands adiabatically, meaning that there is no heat transfer between the system and its surroundings. 3. Isothermal compression: The system compresses isothermally, releasing heat to the reservoir. 4. Adiabatic compression: The system compresses adiabatically, meaning that there is no heat transfer between the system and its surroundings. ### Q3: What is the Carnot efficiency? A3: The Carnot efficiency is a measure of the efficiency of a heat engine. It is defined as the ratio of the work done by the engine to the heat absorbed from the reservoir. The Carnot efficiency is represented by the equation: $\eta = 1 - \frac{T_C}{T_H} </span></p> <p>where <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>η</mi></mrow><annotation encoding="application/x-tex">\eta</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">η</span></span></span></span> is the Carnot efficiency, <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>T</mi><mi>C</mi></msub></mrow><annotation encoding="application/x-tex">T_C</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">T</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.07153em;">C</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> is the temperature of the cold reservoir, and <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>T</mi><mi>H</mi></msub></mrow><annotation encoding="application/x-tex">T_H</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">T</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.08125em;">H</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> is the temperature of the hot reservoir.</p> <h3>Q4: How is the Carnot efficiency used in power plants?</h3> <p>A4: The Carnot efficiency is used to calculate the maximum possible efficiency of a power plant. It is used to determine the ratio of the heat absorbed from the reservoir to the work done by the engine.</p> <h3>Q5: What is the significance of the Carnot cycle in power plants?</h3> <p>A5: The Carnot cycle is significant in power plants because it provides an upper limit to the efficiency of any heat engine. It helps to determine the maximum possible efficiency of a power plant and to design more efficient power plants.</p> <h3>Q6: Can the Carnot cycle be used in real-world power plants?</h3> <p>A6: The Carnot cycle is a theoretical cycle, and it is not possible to achieve 100% efficiency in real-world power plants. However, the Carnot cycle can be used as a reference to design more efficient power plants.</p> <h3>Q7: What are some of the limitations of the Carnot cycle?</h3> <p>A7: Some of the limitations of the Carnot cycle include:</p> <ul> <li>It is a theoretical cycle and not achievable in real-world power plants.</li> <li>It assumes that the heat transfer is reversible, which is not possible in real-world power plants.</li> <li>It assumes that the system is in thermal equilibrium, which is not possible in real-world power plants.</li> </ul> <h3>Q8: Can the Carnot cycle be used in other applications besides power plants?</h3> <p>A8: Yes, the Carnot cycle can be used in other applications besides power plants, such as refrigeration and air conditioning systems.</p> <h2>Conclusion</h2> <p>In this article, we have answered some frequently asked questions related to the Carnot cycle and its application to power plants. We have discussed the stages of the Carnot cycle, the Carnot efficiency, and the significance of the Carnot cycle in power plants. We have also discussed some of the limitations of the Carnot cycle and its applications beyond power plants.</p> <h2>References</h2> <ul> <li>Carnot, S. (1824). Reflections on the Motive Power of Fire and on Machines Fitted to Develop That Power. Bachelier.</li> <li>Feynman, R. P. (1963). The Feynman Lectures on Physics. Addison-Wesley.</li> <li>Halliday, D., Resnick, R., &amp; Walker, J. (2013). Fundamentals of Physics. John Wiley &amp; Sons.</li> </ul>