Question:
A Carnot engine of 50 % efficiency takes heat from a source of 400 K. To change efficiency to 70 % without changing the sink temperature, the new temperature of source should be (nearer to)
A Carnot engine of 50 % efficiency takes heat from a source of 400 K. To change efficiency to 70 % without changing the sink temperature, the new temperature of source should be (nearer to)
Show Hint
In Carnot engine problems, always solve for the unchanged temperature reservoir first!
Updated On: Jun 3, 2026
- 765 K
- 525 K
- 800 K
- 667 K
Show Solution
Verified By Collegedunia
The Correct Option is D
Solution and Explanation
Step 1: Concept
The efficiency ($\eta$) of an ideal Carnot heat engine depends purely on the absolute temperatures of its thermal source ($T_1$) and sink ($T_2$), given by the formula $\eta = 1 - \frac{T_2}{T_1}$.
Step 2: Meaning
Initially, the efficiency is $\eta_1 = 0.50$ and the initial source temperature is $T_1 = 400 \text{ K}$. We can utilize these parameters to first calculate the fixed sink temperature $T_2$.
Step 3: Analysis
From the initial state: $0.50 = 1 - \frac{T_2}{400} \implies \frac{T_2}{400} = 0.50 \implies T_2 = 200 \text{ K}$. For the modified state, we want a target efficiency $\eta_2 = 0.70$ keeping $T_2 = 200 \text{ K}$ unchanged. Let $T_1'$ be the new source temperature. Using the efficiency equation again: $0.70 = 1 - \frac{200}{T_1'} \implies \frac{200}{T_1'} = 0.30 \implies T_1' = \frac{200}{0.30} = \frac{2000}{3} \approx 666.67 \text{ K}$.
Step 4: Conclusion
Rounding to the nearest whole integer yields a source temperature of 667 K.
Final Answer: (D)
The efficiency ($\eta$) of an ideal Carnot heat engine depends purely on the absolute temperatures of its thermal source ($T_1$) and sink ($T_2$), given by the formula $\eta = 1 - \frac{T_2}{T_1}$.
Step 2: Meaning
Initially, the efficiency is $\eta_1 = 0.50$ and the initial source temperature is $T_1 = 400 \text{ K}$. We can utilize these parameters to first calculate the fixed sink temperature $T_2$.
Step 3: Analysis
From the initial state: $0.50 = 1 - \frac{T_2}{400} \implies \frac{T_2}{400} = 0.50 \implies T_2 = 200 \text{ K}$. For the modified state, we want a target efficiency $\eta_2 = 0.70$ keeping $T_2 = 200 \text{ K}$ unchanged. Let $T_1'$ be the new source temperature. Using the efficiency equation again: $0.70 = 1 - \frac{200}{T_1'} \implies \frac{200}{T_1'} = 0.30 \implies T_1' = \frac{200}{0.30} = \frac{2000}{3} \approx 666.67 \text{ K}$.
Step 4: Conclusion
Rounding to the nearest whole integer yields a source temperature of 667 K.
Final Answer: (D)
Was this answer helpful?
0
0
Top AP EAPCET Physics Questions
- A body is projected from the ground at an angle of $\tan^{-1} \left( \frac{8}{7}\right)$ with the horizontal. The ratio of the maximum height attained by it to its range is
- A lens forms real and virtual images of an object, when the object is at $u_{1}$ and $u_{2}$ distances respectively. If the size of the virtual image is double that of the real image, then the focal length of the lens is (take, the magnification of the real image as $m$ )
- Two spheres $P$ and $Q$, each of mass $200\, g$ are attached to a string of length one metre as shown in the figure. The string and the spheres are then whirled in a horizontal circle about $O$ at a constant angular speed. The ratio of the tension in the string between $P$ and $Q$ to that of between $P$ and $O$ is $(P$ is at mid-point of the line joining $O$ and $Q$ )
- The radii of the two columns in a U - tube are r1 and r2. When a liquid of density p (angle of contact is $ 0{}^\circ $ ) is filled in it, the level difference of the liquid in the two arms is h. The surface tension of the liquid is: (g = acceleration due to gravity)
- A rigid metallic sphere is spinning around its own axis in the absence of external torque. If the temperature is raised, its volume increases by $9 \%$. The change in its angular speed is
View More Questions
Top AP EAPCET Thermodynamics Questions
- A solid sphere at a temperature of 400 K radiates a power $P$. If the radius of the sphere is halved and its absolute temperature is doubled, then the power radiated by it is
- When a polyatomic gas is heated at constant pressure, the percentage of heat given to the gas that is converted into external work is
(Ratio of the specific heat capacities of the gas = $\frac{4}{3}$) - The thermodynamic process in which the change in internal energy of the system becomes zero is
- The efficiency of a Carnot engine operating between the temperatures 1000 K and 300 K is
- A closed vessel contains a gas at a pressure $P$. If 50% of the mass of the gas is removed and rms speed of the gas molecules is increased by 20%, then the pressure of the remaining gas is
View More Questions
Top AP EAPCET Questions
- A body is projected from the ground at an angle of $\tan^{-1} \left( \frac{8}{7}\right)$ with the horizontal. The ratio of the maximum height attained by it to its range is
- The four quantum numbers of the valence electron of potassium are :
- A lens forms real and virtual images of an object, when the object is at $u_{1}$ and $u_{2}$ distances respectively. If the size of the virtual image is double that of the real image, then the focal length of the lens is (take, the magnification of the real image as $m$ )
- Two spheres $P$ and $Q$, each of mass $200\, g$ are attached to a string of length one metre as shown in the figure. The string and the spheres are then whirled in a horizontal circle about $O$ at a constant angular speed. The ratio of the tension in the string between $P$ and $Q$ to that of between $P$ and $O$ is $(P$ is at mid-point of the line joining $O$ and $Q$ )
- The volume of $0.02\, M$ acidified permanganate solution required for complete reaction of $60\, mL$ of $0.01\, MI^{-}$ ion solution to form $I_2$ in $mL$ is
View More Questions