Question

As part of your job at the patent office, you are asked to evaluate the six designs shown in the figure for innovative new heat engines. (Figure 1)

Part A

Which of the designs violate(s) the first law of thermodynamics?

Give the letter(s) of the design(s) in alphabetical order, without commas or spaces (e.g., ACD).

Answer 1

Concepts and reason

This question is based upon the concept of the first law of thermodynamics.

First law of thermodynamics, heat energy given to the system is equal to the sum of work done and output power. Validate the first law of thermodynamics for engines A to F.

Fundamentals

The expression for first law of thermodynamics is,

$\Delta U = q + W$

Here, $\Delta U$ is the total internal energy, $q$ is heat and $W$ is the work done by the system.

The expression for first law of thermodynamics also can be written as,

$\frac{{{Q_h}}}{t} = \frac{{{Q_c}}}{t} + W$

Here, ${Q_h}$ is the heat at the higher temperature and ${Q_c}$ heat at the cooler temperature and $W$ is the work done by the system.

For engine A,

The expression for first law of thermodynamics is,

$\frac{{{Q_h}}}{t} = \frac{{{Q_c}}}{t} + W$

Substitute $1000{\rm{ W}}$ for $\frac{{{Q_h}}}{t}$ , $600{\rm{ W}}$ for $\frac{{{Q_c}}}{t}$ and $400{\rm{ W}}$ for $W$ in above equation.

$\begin{array}{c}\\1000{\rm{ W}} = 600{\rm{ W}} + 400{\rm{ W}}\\\\1000{\rm{ W}} = 1000{\rm{ W}}\\\end{array}$

Above equation satisfied the first law of thermodynamics because total input is equal to total output.

For engine B,

The expression for first law of thermodynamics is,

$\frac{{{Q_h}}}{t} = \frac{{{Q_c}}}{t} + W$

Substitute $1000{\rm{ W}}$ for $\frac{{{Q_h}}}{t}$ , $0$ for $\frac{{{Q_c}}}{t}$ and $400{\rm{ W}}$ for $W$ in above equation.

$\begin{array}{c}\\1000{\rm{ W}} = 0 + 400{\rm{ W}}\\\\1000{\rm{ W}} = 1000{\rm{ W}}\\\end{array}$

Above equation is also satisfied the first law of thermodynamics.

For engine C,

The expression for first law of thermodynamics is,

$\frac{{{Q_h}}}{t} = \frac{{{Q_c}}}{t} + W$

Substitute $1000{\rm{ W}}$ for $\frac{{{Q_h}}}{t}$ , $600{\rm{ W}}$ for $\frac{{{Q_c}}}{t}$ and $600{\rm{ W}}$ for $W$ in above equation.

$\begin{array}{c}\\1000{\rm{ W}} = 600{\rm{ W}} + 600{\rm{ W}}\\\\1000{\rm{ W}} \ne 1200{\rm{ W}}\\\end{array}$

Total input is not equal to total output so above equation is not satisfied the equation of first law of thermodynamics.

For engine D,

The expression for first law of thermodynamics is,

$\frac{{{Q_h}}}{t} = \frac{{{Q_c}}}{t} + W$

Substitute $1000{\rm{ W}}$ for $\frac{{{Q_h}}}{t}$ , $500{\rm{ W}}$ for $\frac{{{Q_c}}}{t}$ and $500{\rm{ W}}$ for $W$ in above equation.

$\begin{array}{c}\\1000{\rm{ W}} = 500{\rm{ W}} + 500{\rm{ W}}\\\\1000{\rm{ W}} = 1000{\rm{ W}}\\\end{array}$

Above equation satisfied the first law of thermodynamics.

For engine E,

The expression for first law of thermodynamics is,

$\frac{{{Q_h}}}{t} = \frac{{{Q_c}}}{t} + W$

Substitute $1000{\rm{ W}}$ for $\frac{{{Q_h}}}{t}$ , $550{\rm{ W}}$ for $\frac{{{Q_c}}}{t}$ and $450{\rm{ W}}$ for $W$ in above equation.

$\begin{array}{c}\\1000{\rm{ W}} = 550{\rm{ W}} + 450{\rm{ W}}\\\\1000{\rm{ W = }}1200{\rm{ W}}\\\end{array}$

Above equation satisfied the first law of thermodynamics.

For engine F,

The expression for first law of thermodynamics is,

$\frac{{{Q_h}}}{t} = \frac{{{Q_c}}}{t} + W$

Substitute $1000{\rm{ W}}$ for $\frac{{{Q_h}}}{t}$ , $250{\rm{ W}}$ for $\frac{{{Q_c}}}{t}$ and $650{\rm{ W}}$ for $W$ in above equation.

$\begin{array}{c}\\1000{\rm{ W}} = 250{\rm{ W}} + 650{\rm{ W}}\\\\1000{\rm{ W}} \ne 900{\rm{ W}}\\\end{array}$

Above equation is not satisfied the first law of thermodynamics.

From the above equations for the engines A, B, D and E first law of thermodynamics is satisfied. First law of thermodynamics is not satisfied for engines C and F because in these systems total input is not equal to total output.

Ans:

The first law of thermodynamics is not valid for engines C and F.

Answer 2

The design that violates the first law of thermodynamics are:

CF