An inventor claims to have invented four engines, each of which operates between constant-temperature reservoirs at 400 and 300K. Data on each engine, per cycle of operation, are: engine A, ${\mathbf{Q}}_{\mathbf{H}}\mathbf{=}\mathbf{200}\mathbf{J}\mathbf{,}\mathbf{}{\mathbf{Q}}_{\mathbf{L}}\mathbf{=}\mathbf{175}\mathbf{J}$, and W = 40 J; engine B, ${\mathbf{Q}}_{\mathbf{H}}\mathbf{=}\mathbf{500}\mathbf{J}\mathbf{,}\mathbf{}{\mathbf{Q}}_{\mathbf{L}}\mathbf{=}\mathbf{-}\mathbf{200}\mathbf{}\mathbf{J}$,and W = 400 J; engine C, ${\mathbf{Q}}_{\mathbf{H}}\mathbf{=}\mathbf{600}\mathbf{J}\mathbf{,}\mathbf{}{\mathbf{Q}}_{\mathbf{L}}\mathbf{=}\mathbf{-}\mathbf{200}\mathbf{J}$, and W = 400 J; engine D,${\mathbf{Q}}_{\mathbf{H}}\mathbf{=}\mathbf{100}\mathbf{J}\mathbf{,}{\mathbf{Q}}_{\mathbf{L}}\mathbf{=}\mathbf{-}\mathbf{90}\mathbf{J}$, and W = 10J. Of the first and second laws of thermodynamics, which (if either) does each engine violate?

$$\begin{gathered} a)\mathrm{The}\mathrm{lower}\mathrm{temperature}\mathrm{of}\mathrm{the}\mathrm{four}\mathrm{engines}\mathrm{are}{T}_L=300K \\ b)\mathrm{The}\mathrm{higher}\mathrm{temperature}\mathrm{of}\mathrm{the}\mathrm{four}\mathrm{engines}\mathrm{are}{T}_H=400K \\ c)ForengineA:Q_H=200\mathrm{J},Q_L=-175\mathrm{J}\mathrm{and}W=40\mathrm{J} \\ d)ForengineB:Q_H=500\mathrm{J},Q_L=-200\mathrm{J}\mathrm{and}W=400\mathrm{J} \\ e)ForengineC:Q_H=600\mathrm{J},Q_L=-200\mathrm{J}\mathrm{and}W=400\mathrm{J} \\ f)ForengineD:Q_H=100\mathrm{J},Q_L=-90\mathrm{J}\mathrm{and}W=10\mathrm{J} \end{gathered}$$

The term "Carnot efficiency" refers to the highest level of thermal performance that a heat engine is capable of under the Second Law of Thermodynamics. We can find the efficiency of the Carnot engine and all four engines. Then comparing them with Carnot's engine efficiency we can determine whether the second law fails or not. Then, by calculating the difference between heat and work, we can determine whether the first law is violated.

Formulae:

The efficiency of the Carnot cycle, ${\epsilon }_c=1-\frac{T_L}{T_H}\mathrm{or}{\epsilon }_c=\frac{\left|W\right|}{\left|Q_H\right|}$ …(i)

The change in internal energy due to first law of thermodynamics, $∆E_{int}=Q-W$ …(ii)

The net energy of a Carnot cycle, $Q-Q_H-Q_L$ …(iii)

Using the given values in equation (i), we can get the efficiency of the Carnot engine as follows:

$$\begin{gathered} {\epsilon }_c=1-\frac{300\mathrm{K}}{400\mathrm{K}} \\ =0.25 \end{gathered}$$

For engine A , the net energy for engine A is given using equation (iii) as follows:

Q = 200 J - 175 J

=0.25 J

The efficiency of engine A is given using equation (i) as follows:

$$\begin{gathered} {\epsilon }_c=\frac{40\mathrm{J}}{200\mathrm{J}} \\ =0.20 \end{gathered}$$

For engine B, the net energy for engine B is given using equation (iii) as follows:

$$\begin{gathered} Q=500\mathrm{J}-200\mathrm{J} \\ =300\mathrm{J} \end{gathered}$$

The efficiency of engine B is given using equation (i) as follows:

$$\begin{gathered} {\epsilon }_B=\frac{400\mathrm{J}}{500\mathrm{J}} \\ =0.80 \end{gathered}$$

For engine C, the net energy for engine C is given using equation (iii) as follows:

Q = 600 J - 200 J

= 400 J

The efficiency of engine C is given using equation (i) as follows:

$$\begin{gathered} {\epsilon }_c=\frac{400\mathrm{J}}{600\mathrm{J}} \\ =0.67 \end{gathered}$$

For engine D, the net energy for engine D is given using equation (iii) as follows:

Q = 100 J - 90 J

=0.67

The efficiency of engine D is given using equation (i) as follows:

$$\begin{gathered} {\epsilon }_D=\frac{10\mathrm{J}}{100\mathrm{J}} \\ =0.10 \end{gathered}$$

From equation (ii), we can get the change in internal energy as follows:

Q - W = 0

This is the condition for engine not violating the first law of thermodynamics.

For engine A:

Q - W = 25 J - 40 J

= 15 J

= 0

Thus, the engine A violates the first law.

For engine B:

Q - W = 300 J - 400 J

= 100 J

= 0

For engine C:

Q - W = 400 J - 400 J

= 0 J

For engine D:

Q - W = 10 J - 10 J

= 0 J

Thus, the engine B violates the first law.

The efficiency of the engine is not greater than the Carnot engine.

${\epsilon }_c>\epsilon$

This is the condition of not violating the second law of thermodynamics.

The efficiency of engines B and C is greater than the efficiency of the Carnot engine which violates the second law of thermodynamics.

Engines B and C violate the second law of thermodynamics.

Engines A and B violate the first law of thermodynamics, and engines B and C violate the second law of thermodynamics. Engine D does not violate any law.

Thus, engine D obeys both laws.