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$\backslash$sigma $.$ As a result of the operation of the Carnot engines, the temperature of the fluid decreases from T$1$ to T$2.$ The equation $\backslash$omega $=$ QCW

applies here in differential form, wherein $\backslash$eta is defined as

$\backslash$eta $=$ dWdQ

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$\backslash$sigma $\backslash$Delta S

$(2)$ W $=-$ T$\backslash$sigma $\backslash$Delta S

$(3)$ W $=$ T$\backslash$sigma $\backslash$Delta S

$(4)$ W $=$ Q $-$ T$\backslash$sigma $\backslash$Delta S

The correct expression is in option

$($Click to select$)$

$.$

In a particular case, the fluid is an ideal gas, with CP $=(7/2)$R$,$ and the operating temperatures are T$1=520$ K and T$2=400$ K$.$ If T$\backslash$sigma $=300$ K$,$ what is the value of W in J$$mol$1?$ How much heat is discarded to the heat reservoir at T$\backslash$sigma $?$ What is the entropy change of the heat reservoir? What is $$Stotal$?$

W $=$

J$$mol$1$

The amount of heat discarded to the heat reservoir is

J$$mol$1.$

The entropy change of the heat reservoir is

J$$mol$1$K$1.$

$$Stotal $=$

J$$mol$1$K$1$