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]
