The chart shows data for four heat engines.

\begin{tabular}{|l|c|c|}
\hline Heat Engine & [tex]$T _{ h } ( K )$[/tex] & [tex]$T _{ o } ( K )$[/tex] \\
\hline W & 620 & 120 \\
\hline X & 840 & 100 \\
\hline Y & 900 & 300 \\
\hline Z & 500 & 25 \\
\hline
\end{tabular}

Which lists the engines from most efficient to least efficient?

A. Y, X, W, Z

B. [tex]$Z, W, X, Y$[/tex]

C. [tex]$Z, X, W, Y$[/tex]

D. [tex]$Y, W, X, Z$[/tex]



Answer :

To determine the efficiencies of the heat engines and rank them from most efficient to least efficient, we can utilize the Carnot efficiency formula, which is given by:

[tex]\[ \eta = 1 - \frac{T_o}{T_h} \][/tex]

where [tex]\( \eta \)[/tex] is the efficiency, [tex]\( T_o \)[/tex] is the temperature of the cold reservoir, and [tex]\( T_h \)[/tex] is the temperature of the hot reservoir.

Let's calculate the efficiency for each engine:

1. Engine W:
[tex]\[ T_h = 620 \, K, \quad T_o = 120 \, K \][/tex]
[tex]\[ \eta_W = 1 - \frac{120}{620} \approx 1 - 0.1935 \approx 0.8065 \][/tex]

2. Engine X:
[tex]\[ T_h = 840 \, K, \quad T_o = 100 \, K \][/tex]
[tex]\[ \eta_X = 1 - \frac{100}{840} \approx 1 - 0.1190 \approx 0.8810 \][/tex]

3. Engine Z:
[tex]\[ T_h = 900 \, K, \quad T_o = 300 \, K \][/tex]
[tex]\[ \eta_Z = 1 - \frac{300}{900} \approx 1 - 0.3333 \approx 0.6667 \][/tex]

4. Engine Y:
[tex]\[ T_h = 500 \, K, \quad T_o = 25 \, K \][/tex]
[tex]\[ \eta_Y = 1 - \frac{25}{500} \approx 1 - 0.0500 \approx 0.9500 \][/tex]

We have calculated the efficiencies as follows:
- [tex]\(\eta_W \approx 0.8065\)[/tex]
- [tex]\(\eta_X \approx 0.8810\)[/tex]
- [tex]\(\eta_Z \approx 0.6667\)[/tex]
- [tex]\(\eta_Y \approx 0.9500\)[/tex]

Now, let's rank the engines from most efficient to least efficient based on these efficiencies:
1. [tex]\(\eta_Y = 0.9500\)[/tex]
2. [tex]\(\eta_X = 0.8810\)[/tex]
3. [tex]\(\eta_W = 0.8065\)[/tex]
4. [tex]\(\eta_Z = 0.6667\)[/tex]

Thus, the correct order from most efficient to least efficient is:

[tex]\[ Y, X, W, Z \][/tex]

So, the correct answer is:
[tex]\[ \boxed{Y, X, W, Z} \][/tex]