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A Carnot engine takes 500 kcal of heat from a reservoir at 627°C and exhausts it to a sink at 27°C. How much heat was passed to the sink?

A Carnot engine takes 500 kcal of heat from a reservoir at 627°C and exhausts it to a sink at 27°C. How much heat was passed to the sink? Correct Answer <span style='display: inline !important; float: none; background-color: transparent; color: rgb(51, 51, 51); font-family: "Helvetica Neue",Helvetica,Arial,sans-serif; font-size: 14px; font-style: normal; font-variant: normal; font-weight: 400; letter-spacing: normal; orphans: 2; text-align: left; text-decoration: none; text-indent: 0px; text-transform: none; -webkit-text-stroke-width: 0px; white-space: normal; word-spacing: 0px;'>1.67 </span><span style="background-color: transparent; box-sizing: border-box; color: rgb(51, 51, 51); font-family: &amp;quot;Calibri&amp;quot;,sans-serif; font-size: 14.66px; font-style: normal; font-variant: normal; font-weight: 400; letter-spacing: normal; line-height: 16px; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; orphans: 2; text-align: left; text-decoration: none; text-indent: 0px; text-transform: none; -webkit-text-stroke-width: 0px; white-space: normal; word-spacing: 0px;"><font color="#000000" style="box-sizing: border-box;">× </font></span><span style='display: inline !important; float: none; background-color: transparent; color: rgb(51, 51, 51); font-family: "Helvetica Neue",Helvetica,Arial,sans-serif; font-size: 14px; font-style: normal; font-variant: normal; font-weight: 400; letter-spacing: normal; orphans: 2; text-align: left; text-decoration: none; text-indent: 0px; text-transform: none; -webkit-text-stroke-width: 0px; white-space: normal; word-spacing: 0px;'>10</span><sup style="background-color: transparent; box-sizing: border-box; color: rgb(51, 51, 51); font-family: &amp;quot;Helvetica Neue&amp;quot;,Helvetica,Arial,sans-serif; font-size: 8.8px; font-style: normal; font-variant: normal; font-weight: 400; letter-spacing: normal; line-height: 0px; orphans: 2; position: relative; text-align: left; text-decoration: none; text-indent: 0px; text-transform: none; top: -4.4px; vertical-align: baseline; -webkit-text-stroke-width: 0px; white-space: normal; word-spacing: 0px;">5</sup><span style='display: inline !important; float: none; background-color: transparent; color: rgb(51, 51, 51); font-family: "Helvetica Neue",Helvetica,Arial,sans-serif; font-size: 14px; font-style: normal; font-variant: normal; font-weight: 400; letter-spacing: normal; orphans: 2; text-align: left; text-decoration: none; text-indent: 0px; text-transform: none; -webkit-text-stroke-width: 0px; white-space: normal; word-spacing: 0px;'> cal</span>

CONCEPT:

The efficiency of the Carnot cycle (η):

  • It is defined as the ratio of net mechanical work done per cycle the gas (W) to the amount of heat energy absorbed per cycle from the source (Q1) i.e.,

\(\eta = \frac{W}{{{Q_1}}}\;\)

As work is done by the engine per cycle is

W = Q1 – Q2

Where, Q1 = amount of heat energy absorbed per cycle from the source and Q2 = energy absorbed per cycle from the sink.

\( \Rightarrow \eta = \frac{{{Q_1} - {Q_2}}}{{{Q_1}}} = 1 - \frac{{{Q_2}}}{{{Q_1}}}\)

As, \(\frac{{{Q_2}}}{{{Q_1}}} = \frac{{{T_2}}}{{{T_1}}}\)

\( \Rightarrow \eta = 1 - \frac{{{T_2}}}{{{T_1}}}\)

Where T1 = temperature of the sink and T2 = temperature of the source.

CALCULATION:

Given - 

Q1 = 500 kcal = 5 × 105 cal

T1 = 627 + 273 = 900 K

T2 = 27 + 273 = 300 K

To find Q2

We know that,

 \(\frac{Q_2}{Q_1} = \frac{T_2}{T_1}\)

or, Q2 = \(\frac{T_2 Q_1}{T_1}\)

= \(\frac{300 × 5 × 10^5}{900}\)

= 1.67 × 105 cal

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