Question

Two carnot engines a and b have their sources at 327 and 227 and sinks at 127 and 27 ratio of efficiency of a to that of b

Answer 1

Given:

For Carnot engine A:

Temperature of source $\rm ({T_1}_{A})$ = 327 K

Temperature of sink $\rm ({T_2}_{A})$ = 127 K

For Carnot engine B:

Temperature of source $\rm ({T_1}_{B})$ = 227 K

Temperature of sink $\rm ({T_2}_{B})$ = 27 K

To Find:

Ratio of efficiency of A to that of B $\rm \eta_A : \eta_B$

Answer:

Efficiency of Carnot engine:

$\boxed{ \bf{\eta = 1 - \dfrac{T_2}{T_1}}}$

So,

$\rm \implies \dfrac{\eta_A}{\eta_B} = \dfrac{1 - \dfrac{{T_2}_{A}}{{T_1}_{A}}}{1 - \dfrac{{T_2}_{B}}{{T_1}_{B}}} \\ \\ \rm \implies \dfrac{\eta_A}{\eta_B} = \dfrac{1 - \dfrac{127}{327}}{1 - \dfrac{27}{227}} \\ \\ \rm \implies \dfrac{\eta_A}{\eta_B} = \dfrac{ \dfrac{327 - 127}{327}}{ \dfrac{227 - 27}{227}} \\ \\ \rm \implies \dfrac{\eta_A}{\eta_B} = \dfrac{ \dfrac{ \cancel{200}}{327}}{ \dfrac{ \cancel{200}}{227}} \\ \\ \rm \implies \dfrac{\eta_A}{\eta_B} = \dfrac{ \dfrac{1}{327}}{ \dfrac{1}{227}} \\ \\ \rm \implies \dfrac{\eta_A}{\eta_B} = \dfrac{ 227}{ 327} \\ \\ \rm \implies \eta_A : \eta_B = 227 : 327$

$\therefore$ $\boxed{\mathfrak{Ratio \ of \ efficiency \ of \ A \ to \ that \ of \ B \ (\eta_A : \eta_B) = 227 : 327}}$

Answer 2

GIVEN :-

• Tᴡᴏ ᴄᴀʀɴᴏᴛ ᴇɴɢɪɴᴇs $\bf\red{a}$ ᴀɴᴅ $\bf\red{b}$ ʜᴀᴠᴇ ᴛʜᴇɪʀ sᴏᴜʀᴄᴇs ᴀᴛ $\bf\red{327\:K}$ ᴀɴᴅ $\bf\red{227\:K}$ ᴀɴᴅ sɪɴᴋs ᴀᴛ $\bf\red{127\:K}$ ᴀɴᴅ $\bf\red{27\:K}$ .

TO FIND :-

• Rᴀᴛɪᴏ ᴏғ ᴇғғɪᴄɪᴇɴᴄʏ ᴏғ $\bf\red{a}$ ᴛᴏ ᴛʜᴀᴛ ᴏғ $\bf\red{b}$ .

SOLUTION :-

ᴡᴇ ʜᴀᴠᴇ ᴋɴᴏᴡ ᴛʜᴀᴛ,

$\red\checkmark\:\bf\purple{Efficiency\:of\:Carnot\:engine\:(\eta)\:=\:1\:-\:\dfrac{T_2}{T_1}\:}$

Fᴏʀ ᴄᴀʀɴᴏᴛ ᴇɴɢɪɴᴇ "a" :-

$\huge\blue\star$ $\bf\orange{\eta_{a}\:=\:1\:-\:\dfrac{T_2}{T_1}\:}$

ᴡʜᴇʀᴇ,

• $\bf\pink{T_2}$ = 127 K

• $\bf\pink{T_1}$ = 327 K

$\bf{:\implies\:\eta_{a}\:=\:1\:-\:\dfrac{127}{327}\:}$

$\rm{:\implies\:\eta_{a}\:=\:\dfrac{327\:-\:127}{327}\:}$

$\bf\red{:\implies\:\eta_{a}\:=\:\dfrac{200}{327}\:}$

Fᴏʀ ᴄᴀʀɴᴏᴛ ᴇɴɢɪɴᴇ "b" :-

$\huge\pink\star$ $\bf\green{\eta_{b}\:=\:1\:-\:\dfrac{T_2}{T_1}\:}$

ᴡʜᴇʀᴇ,

• $\bf\pink{T_2}$ = 27 K

• $\bf\pink{T_1}$ = 227 K

$\bf{:\implies\:\eta_{b}\:=\:1\:-\:\dfrac{27}{227}\:}$

$\rm{:\implies\:\eta_{b}\:=\:\dfrac{227\:-\:27}{227}\:}$

$\bf\red{:\implies\:\eta_{b}\:=\:\dfrac{200}{227}\:}$

Nᴏᴡ,

$\bf{:\implies\:\dfrac{\eta_{a}}{\eta_{b}}\:=\:\dfrac{200/327}{200/227}\:}$

$\rm{:\implies\:\:\dfrac{\eta_{a}}{\eta_{b}}\:=\:\dfrac{\cancel{200}}{327}\:\times{\dfrac{227}{\cancel{200}}}\:}$

$\bf{:\implies\:\dfrac{\eta_{a}}{\eta_{b}}\:=\:\dfrac{227}{327}\:}$

$\bf\blue{:\implies\:\eta_{a}\::\:\eta_{b}\:=\:227\::327\:}$

$\huge\red\therefore$ Rᴀᴛɪᴏ ᴏғ ᴇғғɪᴄɪᴇɴᴄʏ ᴏғ $\bf\red{a}$ ᴛᴏ ᴛʜᴀᴛ ᴏғ $\bf\red{b}$ ɪs "227 : 327" .