Imagine that a stream of fluid in steady$-$state flow serves as a heat source for an infinite set of Carnot engines, each of which absorbs a

differential amount of heat from the fluid, causing its temperature to decrease by a differential amount, and each of which rejects a

differential amount of heat to a heat reservoir at temperature $T_{}.$ As a result of the operation of the Carnot engines, the temperature of

the fluid decreases from $T_1$ to $T_2.$ The equation $=\frac{Q_C}{W}$ applies here in differential form, wherein $$ is defined as

$=\frac{dW}{dQ}$

where $Q$ is the heat transfer with respect to the flowing fluid. Identify the correct expression for the total work of the Carnot engines.

$S$ and $Q$ both refer to the fluid.

$(1)W=Q+T_{}S$

$(2)W=-T_{}S$

$(3)W=T_{}S$

$(4)W=Q-T_{}S$

The correct expression is in option

In a particular case, the fluid is an ideal gas, with $C_P=(\frac{7}{2})R,$ and the operating temperatures are $T_1=520K$ and $T_2=400K.$ If $T_{}=$

$300K,$ what is the value of $W$ in $J*mo{l}^{-1}?$ How much heat is discarded to the heat reservoir at $T_{}?$ What is the entropy change of the

heat reservoir? What is $S_{\text{t}\text{o}\text{t}\text{a}\text{l}}?$

$W=,J*mo{l}^{-1}$

The amount of heat discarded to the heat reservoir is J$.$mol ${?}^{-1}.$

The entropy change of the heat reservoir is

The entropy change $of$ the heat reservoir $isJ*mo{l}^{-1}*K^{-1}.$

$S_{\text{t}\text{o}\text{t}\text{a}\text{l}}=,J*mo{l}^{-1}*K^{-1}$