Question

A Carnot engine , whose efficiency is 40% , takes in heat from a source maintained at a temperature of 500 K . It is desired to have an engine of efficiency 60%. Then , the intake temperature for the same exhaust ( sink ) temperature must be :
( 1 ) 750 K
( 2 ) 600 K
( 3 ) Efficiency of Carnot engine cannot be made larger than 50 %
( 4 ) 1200 K​

Answer 1

$\Large\bold{\underline{\underline{Answer:-}}}$

$\bf\huge\boxed{750\:K}$

$\rule{200}{4}$

$\bf\small\bold{\underline{\underline{Step-By-Step\:Explanation:-}}}$

$\bold{\Rightarrow P = 220 \times 0.50 }$

$\green{\texttt{Problems based on Carnot Engine }}$

$Given-\\ \eta_1 =0.4\:\:\:\:\:T_1=500\:k\:\:\:\:\:\:\:T_2=? =x(let) \\\\\eta_2=0.6\:\:\:\:\:T'_1=? \:\:\:\:\:\:T'_2= x(same) \\\\case-1\\\Rightarrow\eta_1=1-\frac{T_2}{T_1}\\\Rightarrow0.4=1-\frac{x}{500}\\\Rightarrow\:x=300\:k\\\\case-2\\\Rightarrow\eta_2=1-\frac{T'_2}{T'_1}\\\Rightarrow0. 6=1-\frac{300}{T'_1}\\\Rightarrow\:T'_1=750\:k$

Answer 2

$\huge{\mathfrak{\underline{\underline{\red{Answer :-}}}}}$

$\large{\mathrm{\gray{\underline{Given:-}}}}$

$\bf{n_{1} \: = \: \frac{40}{100} \: = \: 0.4}$

$\bf{T_{1} \: = \: 500 K}$

$\bf{Let \: T_{2} \: be \: a. }$

$\bf{n_{2} \: = \: \frac{60}{100} \: = \: 0.6}$

$\bf{Let \: T_{3} \: be \: b}$

$\bf{Let \: T_{4} \: be \: a. }$

______________________

$\large{\mathrm{\gray{\underline{To \: Find :-}}}}$

$\bf{{\implies}T_{3}}$

$\bf{{\implies}T_{2} \: and \: T_{4}}$

_______________________

$\large{\mathrm{\gray{\underline{Solution :-}}}}$

$\bf{\bold{Firstly,\: we \: will \: find \: T_{2} \: and \: T_{4}}}$

Using Formula :-

$\bf{\boxed{\boxed{\pink{n_{1} \: =\: \frac{T_{1}}{T_{2}}}}}}$

____________[Put values]

⇒ 0.4 = 1 - a/500

Taking LCM

⇒ 0.4 = 500 - a / 500

⇒ 0.4 * 500 = 500 - a

⇒ 200 = 500 - a

⇒200 - 500 = -a

⇒ - 300 = -a

⇒ a = 300 K

$\huge{\boxed{\boxed{\green{a \: = \: 300 \: K}}}}$

$\rule{200}{2}$

$\bf{\bold{ Now, \: we \: will \: find \: T_{3}}}$

Using Formula :-

$\bf{\boxed{\boxed{\pink{n_{2} \: =\: \frac{T_{4}}{T_{3}}}}}}$

____________[Put values]

⇒ 0.6 = 1 - 300/b

⇒ b = 750 K

$\huge{\boxed{\boxed{\green{ b \: = \: 750 \: K}}}}$