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Question: A Carnot engine takes 100 calories of heat from a reservoir at \[{427^0}C\]and performs 60 Kcal of w...

A Carnot engine takes 100 calories of heat from a reservoir at 4270C{427^0}Cand performs 60 Kcal of work. Calculate the amount of heat absorbed in joule.

A. 10 J
B. 20 J
C. 68 J
D. None of the above

Explanation

Solution

In this question, we need to determine the amount of heat absorbed by the Carnot engine for the useful work of 600 Kcal to be done such that it takes 100 calories of heat from the reservoir. For this, we will use the relation between the heat exchange from the Carnot engine and the work is done, which is given as W=Q1−Q2W = {Q_1} - {Q_2}.

Complete step by step answer: The difference in the heat exchange from the reservoir in the Carnot engine results in the work done by the Carnot engine. Mathematically, W=Q1−Q2W = {Q_1} - {Q_2} where, WW is the work done by the Carnot engine, Q1{Q_1} is the heat taken by the Carnot engine from the reservoir and Q2{Q_2} is the heat absorbed by the Carnot engine. Here, W=60Kcal and Q1=100calW = 60Kcal{\text{ and }}{Q_1} = 100cal.

So, substitute W=60Kcal and Q1=100calW = 60Kcal{\text{ and }}{Q_1} = 100cal in the formula W=Q1−Q2W = {Q_1} - {Q_2} to determine the heat absorbed by the Carnot engine in calories.

W=Q1−Q2 60Kcal=100Cal−Q2 Q2=100cal−60Kcal Q2=100cal−60000cal Q2=−59900cal−−−−(i)  W = {Q_1} - {Q_2} \\\ 60Kcal = 100Cal - {Q_2} \\\ {Q_2} = 100cal - 60Kcal \\\ {Q_2} = 100cal - 60000cal \\\ {Q_2} = - 59900cal - - - - (i) \\\

Now, one calorie of heat is equivalent to 4.184 joules, i.e., 1cal=4.184joules1cal = 4.184joules

Hence, the result in equation (i) can be converted into joules as.

\-59900cal=−59900×4.184joules =−250621.6joules  \- 59900cal = - 59900 \times 4.184joules \\\ = - 250621.6joules \\\

Hence, the amount of heat absorbed by the Carnot engine from the reservoir is -250621.6 Joules, which is also equivalent to -250.62 Kjoules. Hence, none of the given options are correct.

Option D is correct.

Note: Students must be careful while using the formula W=Q1−Q2W = {Q_1} - {Q_2} sign convention plays an important role here. Q1{Q_1} is the heat taken by the Carnot engine from the reservoir (which should always be positive with respect to the reservoir) and Q2{Q_2} is the heat absorbed by the Carnot engine (which should always be negative with respect to the reservoir).