PRELIMINARY STAGE 2026
Konongo Odumasi Senior High School: 45 points (Winner, qualified to one-eighth stage)
Our Lady of Mount Carmel Girls' Senior High School: 31 points
Jaben Senior High School: 28 points
QUESTION
Find the thermal efficiency of a Carnot engine that operates between the given pair of bath temperatures.
1. 25°C and 95°C.
ANSWER: 19.0%
SOLUTION 1:
$T_c = 25 + 273.15$
$T_c = 298.15 \text{ K}$
$T_h = 95 + 273.15$
$T_h = 368.15 \text{ K}$
$\eta = 1 - \dfrac{T_c}{T_h}$
$\eta = 1 - \dfrac{298.15}{368.15}$
$\eta \approx 0.1901$
$\eta \approx 19.0\%$
SOLUTION 2:
$T_c = 25 + 273$
$T_c = 298 \text{ K}$
$T_h = 95 + 273$
$T_h = 368 \text{ K}$
$\eta = 1 - \dfrac{T_c}{T_h}$
$\eta = 1 - \dfrac{298}{368}$
$\eta \approx 0.1902$
$\eta \approx 19.0\%$
2. 15°C and 125°C.
ANSWER: 27.6%
SOLUTION 1:
$T_c = 15 + 273.15$
$T_c = 288.15 \text{ K}$
$T_h = 125 + 273.15$
$T_h = 398.15 \text{ K}$
$\eta = 1 - \dfrac{T_c}{T_h}$
$\eta = 1 - \dfrac{288.15}{398.15}$
$\eta \approx 0.2763$
$\eta \approx 27.6\%$
SOLUTION 2:
$T_c = 15 + 273$
$T_c = 288 \text{ K}$
$T_h = 125 + 273$
$T_h = 398 \text{ K}$
$\eta = 1 - \dfrac{T_c}{T_h}$
$\eta = 1 - \dfrac{288}{398}$
$\eta \approx 0.2764$
$\eta \approx 27.6\%$
3. 27°C and 82°C.
ANSWER: 15.5%
SOLUTION 1:
$T_c = 27 + 273.15$
$T_c = 300.15 \text{ K}$
$T_h = 82 + 273.15$
$T_h = 355.15 \text{ K}$
$\eta = 1 - \dfrac{T_c}{T_h}$
$\eta = 1 - \dfrac{300.15}{355.15}$
$\eta \approx 0.1549$
$\eta \approx 15.5\%$
SOLUTION 2:
$T_c = 27 + 273$
$T_c = 300 \text{ K}$
$T_h = 82 + 273$
$T_h = 355 \text{ K}$
$\eta = 1 - \dfrac{T_c}{T_h}$
$\eta = 1 - \dfrac{300}{355}$
$\eta \approx 0.1549$
$\eta \approx 15.5\%$
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PRACTICE QUESTIONS
1. QUESTION: Find the thermal efficiency of a Carnot engine that operates between baths at $3.00\times10^{2}$ K and $4.00\times10^{2}$ K.
ANSWER: 25.0%
SOLUTION:
$\eta = 1 - \dfrac{T_c}{T_h}$
$\dfrac{T_c}{T_h} = \dfrac{3.00\times10^{2}}{4.00\times10^{2}}$
$\dfrac{T_c}{T_h} = \dfrac{3\times10^{2}}{4\times10^{2}}$
$\dfrac{T_c}{T_h} = \dfrac{3}{4}$
$\dfrac{T_c}{T_h} = 0.750$
$\eta = 1 - 0.750$
$\eta = 0.250$
$\eta = 25.0\%$
2. QUESTION: Find the thermal efficiency of a Carnot engine that operates between baths at $2.00\times10^{2}$ K and $5.00\times10^{2}$ K.
ANSWER: 60.0%
SOLUTION:
$\eta = 1 - \dfrac{T_c}{T_h}$
$\dfrac{T_c}{T_h} = \dfrac{2.00\times10^{2}}{5.00\times10^{2}}$
$\dfrac{T_c}{T_h} = \dfrac{2\times10^{2}}{5\times10^{2}}$
$\dfrac{T_c}{T_h} = \dfrac{2}{5}$
$\dfrac{T_c}{T_h} = 0.400$
$\eta = 1 - 0.400$
$\eta = 0.600$
$\eta = 60.0\%$
3. QUESTION: Find the thermal efficiency of a Carnot engine that operates between baths at $3.00\times10^{2}$ K and $6.00\times10^{2}$ K.
ANSWER: 50.0%
SOLUTION:
$\eta = 1 - \dfrac{T_c}{T_h}$
$\dfrac{T_c}{T_h} = \dfrac{3.00\times10^{2}}{6.00\times10^{2}}$
$\dfrac{T_c}{T_h} = \dfrac{3\times10^{2}}{6\times10^{2}}$
$\dfrac{T_c}{T_h} = \dfrac{1}{2}$
$\dfrac{T_c}{T_h} = 0.500$
$\eta = 1 - 0.500$
$\eta = 0.500$
$\eta = 50.0\%$
4. QUESTION: Find the thermal efficiency of a Carnot engine that operates between baths at $3.00\times10^{2}$ K and $8.00\times10^{2}$ K.
ANSWER: 62.5%
SOLUTION:
$\eta = 1 - \dfrac{T_c}{T_h}$
$\dfrac{T_c}{T_h} = \dfrac{3.00\times10^{2}}{8.00\times10^{2}}$
$\dfrac{T_c}{T_h} = \dfrac{3\times10^{2}}{8\times10^{2}}$
$\dfrac{T_c}{T_h} = \dfrac{3}{8}$
$\dfrac{T_c}{T_h} = 0.375$
$\eta = 1 - 0.375$
$\eta = 0.625$
$\eta = 62.5\%$
5. QUESTION: Find the thermal efficiency of a Carnot engine that operates between baths at $4.00\times10^{2}$ K and $5.00\times10^{2}$ K.
ANSWER: 20.0%
SOLUTION:
$\eta = 1 - \dfrac{T_c}{T_h}$
$\dfrac{T_c}{T_h} = \dfrac{4.00\times10^{2}}{5.00\times10^{2}}$
$\dfrac{T_c}{T_h} = \dfrac{4\times10^{2}}{5\times10^{2}}$
$\dfrac{T_c}{T_h} = \dfrac{4}{5}$
$\dfrac{T_c}{T_h} = 0.800$
$\eta = 1 - 0.800$
$\eta = 0.200$
$\eta = 20.0\%$
6. QUESTION: Find the thermal efficiency of a Carnot engine that operates between baths at $5.00\times10^{2}$ K and $8.00\times10^{2}$ K.
ANSWER: 37.5%
SOLUTION:
$\eta = 1 - \dfrac{T_c}{T_h}$
$\dfrac{T_c}{T_h} = \dfrac{5.00\times10^{2}}{8.00\times10^{2}}$
$\dfrac{T_c}{T_h} = \dfrac{5\times10^{2}}{8\times10^{2}}$
$\dfrac{T_c}{T_h} = \dfrac{5}{8}$
$\dfrac{T_c}{T_h} = 0.625$
$\eta = 1 - 0.625$
$\eta = 0.375$
$\eta = 37.5\%$
7. QUESTION: Find the thermal efficiency of a Carnot engine that operates between baths at $3.00\times10^{2}$ K and $1.00\times10^{3}$ K.
ANSWER: 70.0%
SOLUTION:
$\eta = 1 - \dfrac{T_c}{T_h}$
$\dfrac{T_c}{T_h} = \dfrac{3.00\times10^{2}}{1.00\times10^{3}}$
$\dfrac{T_c}{T_h} = \dfrac{3\times10^{2}}{10^{3}}$
$\dfrac{T_c}{T_h} = 0.300$
$\eta = 1 - 0.300$
$\eta = 0.700$
$\eta = 70.0\%$
8. QUESTION: Find the thermal efficiency of a Carnot engine that operates between baths at $2.50\times10^{2}$ K and $4.00\times10^{2}$ K.
ANSWER: 37.5%
SOLUTION:
$\eta = 1 - \dfrac{T_c}{T_h}$
$\dfrac{T_c}{T_h} = \dfrac{2.50\times10^{2}}{4.00\times10^{2}}$
$\dfrac{T_c}{T_h} = \dfrac{25\times10^{1}}{4\times10^{2}}$
$\dfrac{T_c}{T_h} = \dfrac{25}{4} \times10^{-1}$
$\dfrac{T_c}{T_h} = 0.625$
$\eta = 1 - 0.625$
$\eta = 0.375$
$\eta = 37.5\%$
9. QUESTION: Find the thermal efficiency of a Carnot engine that operates between baths at $4.00\times10^{2}$ K and $1.60\times10^{3}$ K.
ANSWER: 75.0%
SOLUTION:
$\eta = 1 - \dfrac{T_c}{T_h}$
$\dfrac{T_c}{T_h} = \dfrac{4.00\times10^{2}}{1.60\times10^{3}}$
$\dfrac{T_c}{T_h} = \dfrac{4\times10^{2}}{16\times10^{2}}$
$\dfrac{T_c}{T_h} = \dfrac{1}{4}$
$\dfrac{T_c}{T_h} = 0.250$
$\eta = 1 - 0.250$
$\eta = 0.750$
$\eta = 75.0\%$