The Fahrenheit and Kelvin scales of temperature will have the same reading at a temperature of:
The Fahrenheit and Kelvin scales of temperature will have the same reading at a temperature of:
-40°F
313°F
574.6°F
732.7°F
The Correct Option is C
Approach Solution - 1
To solve the problem, we need to find the temperature at which the Fahrenheit and Kelvin scales show the same numerical value.
1. Understanding the Relation Between Fahrenheit and Kelvin:
The conversion formulas between Fahrenheit (°F), Celsius (°C), and Kelvin (K) are:
$ T_{K} = T_{C} + 273.15 $
$ T_{F} = \frac{9}{5} T_{C} + 32 $
2. Set Kelvin Equal to Fahrenheit:
We are told that Fahrenheit and Kelvin temperatures are numerically equal, i.e.,
$ T_{K} = T_{F} $
Using the conversion of °F to °C and then to K:
Let $T_C$ be the Celsius temperature.
Then $T_F = \frac{9}{5}T_C + 32$ and $T_K = T_C + 273.15$
3. Set the Equations Equal:
$ \frac{9}{5}T_C + 32 = T_C + 273.15 $
4. Solve for $T_C$:
$ \frac{9}{5}T_C - T_C = 273.15 - 32 $
$ \left(\frac{9}{5} - 1\right) T_C = 241.15 $
$ \frac{4}{5} T_C = 241.15 $
$ T_C = \frac{241.15 \times 5}{4} = 301.4375^\circ C $
5. Convert to Fahrenheit or Kelvin:
$ T_F = \frac{9}{5} \times 301.4375 + 32 = 574.6^\circ F $
$ T_K = 301.4375 + 273.15 = 574.6 \, K $
Final Answer:
The Fahrenheit and Kelvin scales show the same reading at 574.6°F.
Approach Solution -2
The correct option is: (C) : 574.6°F.
To find the temperature at which the Fahrenheit and Kelvin scales have the same reading, we can use the conversion formulas between Fahrenheit (F) and Kelvin (K) temperatures:
F = (9/5)K - 459.67
Where F is the temperature in Fahrenheit and K is the temperature in Kelvin.
Setting the two scales equal to each other:
(9/5)K - 459.67 = K
Now, solve for K:
(9/5)K - K = 459.67
Simplify the equation:
(9/5 - 1)K = 459.67
(4/5)K = 459.67
Now, solve for K:
K = (5/4) * 459.67 ≈ 574.5875
So, at a temperature of approximately 574.5875 Kelvin, the Fahrenheit and Kelvin scales will have the same reading.
Now, convert this temperature to Fahrenheit using the conversion formula:
F = (9/5)K - 459.67
F = (9/5) * 574.5875 - 459.67 ≈ 574.6°F
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Concepts Used:
Thermodynamics
Thermodynamics in physics is a branch that deals with heat, work and temperature, and their relation to energy, radiation and physical properties of matter.
Important Terms
System
A thermodynamic system is a specific portion of matter with a definite boundary on which our attention is focused. The system boundary may be real or imaginary, fixed or deformable.
There are three types of systems:
- Isolated System – An isolated system cannot exchange both energy and mass with its surroundings. The universe is considered an isolated system.
- Closed System – Across the boundary of the closed system, the transfer of energy takes place but the transfer of mass doesn’t take place. Refrigerators and compression of gas in the piston-cylinder assembly are examples of closed systems.
- Open System – In an open system, the mass and energy both may be transferred between the system and surroundings. A steam turbine is an example of an open system.
Thermodynamic Process
A system undergoes a thermodynamic process when there is some energetic change within the system that is associated with changes in pressure, volume and internal energy.
There are four types of thermodynamic process that have their unique properties, and they are:
- Adiabatic Process – A process in which no heat transfer takes place.
- Isochoric Process – A thermodynamic process taking place at constant volume is known as the isochoric process.
- Isobaric Process – A process in which no change in pressure occurs.
- Isothermal Process – A process in which no change in temperature occurs.
Laws of Thermodynamics
Zeroth Law of Thermodynamics
The Zeroth law of thermodynamics states that if two bodies are individually in equilibrium with a separate third body, then the first two bodies are also in thermal equilibrium with each other.
First Law of Thermodynamics
The First law of thermodynamics is a version of the law of conservation of energy, adapted for thermodynamic processes, distinguishing three kinds of transfer of energy, as heat, as thermodynamic work, and as energy associated with matter transfer, and relating them to a function of a body's state, called internal energy.
Second Law of Thermodynamics
The Second law of thermodynamics is a physical law of thermodynamics about heat and loss in its conversion.
Third Law of Thermodynamics
Third law of thermodynamics states, regarding the properties of closed systems in thermodynamic equilibrium: The entropy of a system approaches a constant value when its temperature approaches absolute zero.