Match the temperature of a black body given in L-I with an appropriate statement in L-II and choose the correct option. (given: wien is constant as \(2.9\times10^{-3}\) m.k. \(\frac{hc}{e}=1.24\times10^{-6}\,v.m\) List-I List-II (P) 2000 K (1) The radiation at peek wavelength can lead to the emission of photo e\(^-\) from a metal of work function 4 eV (Q) 3000 K (2) The radiation at peek wavelength is visible to the human eye (R) 5000K (3) The radiation at peek emission wavelength will result in the widest central maximum of a single slit diffraction (S) 10000K (4) The power emitted per unit area is \(\frac{1}{16}\) of that emitted by a black body at a temperature of 6000K. (5) The radiation at peak emission wavelength can be used to image human bone.
| List-I | List-II | ||
| (P) | 2000 K | (1) | The radiation at peek wavelength can lead to the emission of photo e\(^-\) from a metal of work function 4 eV |
| (Q) | 3000 K | (2) | The radiation at peek wavelength is visible to the human eye |
| (R) | 5000K | (3) | The radiation at peek emission wavelength will result in the widest central maximum of a single slit diffraction |
| (S) | 10000K | (4) | The power emitted per unit area is \(\frac{1}{16}\) of that emitted by a black body at a temperature of 6000K. |
| (5) | The radiation at peak emission wavelength can be used to image human bone. |
Approach Solution - 1
The correct answer is : P\(\rightarrow\)3; Q\(\rightarrow\)4; R\(\rightarrow\)2; S\(\rightarrow\)1
\(\lambda\times T=b\)
\(\lambda=\frac{b}{T}\)
\(E=\frac{hc}{\lambda}=\frac{hcT}{b}\)
\(E=(\frac{hc}{eb})\times T\,eb\)
\(E=\frac{1.24\times10^{-6}}{2.9\times10^{-3}}\times T\,ev\)
\(E=(0.428\times 10^{-3}\times T)\,\,eV\)
(P) T = 2000K \(\Rightarrow\) E = 0.856eV
(Q) T = 3000K \(\Rightarrow\) E = 1.284eV
(R) T = 5000K \(\Rightarrow\) E=2.14eV
(S) T = 10000K \(\Rightarrow\) E=4.28eV
Approach Solution -2
\[ \lambda_m T = b \]
For option (P) the temperature is minimum, hence \(\lambda_m\) will be maximum.
\[ \lambda_m = \frac{b}{T} = \frac{2.9 \times 10^{-3}}{2000} = 1.45 \times 10^{-6} \, \text{m} = 1450 \, \text{nm} \]
\[ \sin \theta = \frac{\lambda}{d} \]
\[ 2\theta = \text{width of central maximum} \]
\[\Rightarrow \text{width} \propto \lambda \]
\(\Rightarrow\) maximum width for (P)
\[ P \rightarrow 3 \]
For option (Q), \(T = 3000\)
\[ \lambda_m = \frac{b}{T} = \frac{2.9 \times 10^{-3}}{3000} \]
\[ \lambda_m = 0.96 \times 10^{-6} = 966.6 \, \text{nm} \]
\[ P_{3000} = 6A(3000)^4 \]
\[ P_{6000} = 6A(6000)^4 \]
\[ \frac{P_{3000}}{P_{6000}} = \left( \frac{1}{2} \right)^4 = \frac{1}{16} \]
\[ P_{3000} = \frac{1}{16} P_{6000} \]
\[ Q \rightarrow 4 \]
For (R), \(T = 5000 \, \text{K}\)
\[ \lambda_m = \frac{2.9 \times 10^{-3}}{5 \times 10^{3}} = 0.58 \times 10^{-6} = 580 \, \text{nm} \]
\(\Rightarrow\) Visible to human eyes
\[ R \rightarrow 2 \]
For (S), \(T = 10,000 \, \text{K}\)
\[ \lambda_m T = b \Rightarrow \lambda_m = \frac{2.9 \times 10^{-3}}{10,000} = 290 \, \text{nm} \]
\[ \phi = 4 \, \text{eV} \]
\[ \lambda_m = \frac{1240}{4} = 310 \, \text{nm} \]
To emit photoelectron
\[ \lambda_0 < \lambda_T \]
\(\Rightarrow S \rightarrow 1\)
The correct Answer is : P\(\rightarrow\)3; Q\(\rightarrow\)4; R\(\rightarrow\)2; S\(\rightarrow\)1
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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.