Sensible heat vs Latent heat

The main differentiating point for the sensible heat vs latent heat is that the latent heat is responsible for the change in phase of the substance while the sensible heat is responsible for the change in temperature of the substance.

Latent heat (Non sensible heat):

The amount of heat required for the change in phase of the substance is known as Latent heat.
The latent heat can be classified into three types:

A) Latent heat of fusion/melting:

The amount of heat responsible for the change in phase of the substance from solid to liquid or from liquid to solid is known as latent heat of fusion/melting.

Example: The amount of heat required for the melting of the ice is the latent heat of melting and the heat rejected by the water for the formation of the ice is the latent heat of fusion.

B) Latent heat of vaporization/condensation:

The amount of heat responsible for the change in phase of the substance from liquid to vapor or from vapor to liquid is known as latent heat of vaporization.

Example: The amount of heat required for vaporization of the water is the latent heat of vaporization and the heat rejected by the steam for the condensation is the latent heat of vaporization for the water.

C] Latent heat of sublimation:

The amount of heat responsible for the change in phase of a substance from solid-state to directly into vapor state is known as latent heat of sublimation.

Example: Sublimation of camphor.

Sensible heat:

The heat which is responsible for the change in temperature of the substance is known as Sensible heat. The sensible heat can be sensed manually because of the rise in the temperature of the object. During the exchange of the sensible heat, the object doesn’t change its phase.

Example: The heat required for the heating or cooling of the water.

Difference between Sensible heat and Latent heat:

Sr. No.	Latent heat (Non-Sensible heat)	Sensible heat
1	Latent heat is not responsible for the raising or lowering of the temperature.	Sensible heat is responsible for the raising or lowering of the temperature.
2	The latent heat is responsible for the change in phase of the substance (Solid to liquid, liquid to vapor, and vice versa).	The sensible heat is not responsible for the change in the phase of the substance.
3	During the latent heat transfer, the substance is under phase transition, and the temperature is constant.	During the sensible heat transfer, the temperature of the substance changes without changing the phase of the substance.
4	Example: Heating or the cooling of the water, etc.	Example: Melting of ice, Vaporization or water, etc.

Wien’s displacement law: Statement, Derivation, Formula

What is Wien’s displacement law?

Wien’s displacement law states that the wavelength with the peak emissive power is inversely proportional to the temperature of the black body. This law gives the relation between the temperature of the radiating black body and peak wavelength (wavelength with peak emissive power, ${\lambda }_m$).

The wavelength at which the black body emits maximum monochromatic emissive power is decreases with an increase in the absolute temperature (T) of the black body.

The German physicist Wilhelm Wien found a relationship between these two terms that says,

${\lambda }_m\propto \frac{1}{T}$

${\lambda }_m=\frac{B}{T}$

$T{\lambda }_m=B$

Where, B = 2.897 x 10⁻³ m.K

B is the proportionality constant and it has SI unit of m.K.

From the above equation, Wien’s displacement law can also be stated as the product of ${\lambda }_m$ and T is constant.

Explanation:-

The bodies that are hotter than their surrounding radiates the energy to the surrounding with different wavelengths. This total radiating energy is distributed among these different wavelengths. Thus the energy associated with all wavelengths (Emissive power) varies from each other.

The wavelength that has maximum monochromatic emissive power is considered as ${\lambda }_{\text{max}}$.

The below figure shows the distribution of emissive power among different wavelengths emitted by the blackbody at constant temperature $T_1$.

As shown in the above figure, at a particular point the value of the emissive power is maximum. The wavelength corresponding to the maximum emissive power is the ${\lambda }_{\text{max}}$ or ${\lambda }_{m_1}$.

If we raise the temperature of the black body to $T_2$, then the graph becomes as follows,

Thus at temperature $T_2$ (where $T_2$>$T_1$), peak energy is emitted at a shorter wavelength ${\lambda }_{m_2}$ (${\lambda }_{m_2}$<${\lambda }_{m_1}$).

Similarly, for the different temperatures, the curves of the emissive power distribution are as follows. ($T_4$ > $T_3$ > $T_2$ > $T_1$)

From the above graph, it is observed that as the temperature of the body increases, the wavelength with the peak emissive power displaces toward the right.

The physicist Wilhelm Wien found the relationship between the absolute temperature (T) and peak wavelength (${\lambda }_m$) that is the product of T and its respective value of the ${\lambda }_m$ remains constant.

$T{\lambda }_m$ = Constant

Where,
${\lambda }_m$ = Wavelength at which maximum energy is emitted
T = Absolute temperature of black body

Wien’s displacement law derivation:

As per the plank’s law, the monochromatic emissive power of the blackbody is given by,

${\left(E_{\lambda }\right)}_b$ = $\frac{C_1{\lambda }^{-5}}{e^{\frac{C_2}{\lambda T}}-1}$

Where,
$C_1$ = 3.742 x ${10}^8$ W.µm⁴/m²
$C_2$ = 0.014388 mK

At constant temperature (T), the monochromatic emissive power of blackbody [${\left(E_{\lambda }\right)}_b$] becomes maximum when,

$\frac{d}{d\lambda }{\left(E_{\lambda }\right)}_b$ = 0

$\frac{d}{d\lambda }\left[\frac{C_1{\lambda }^{-5}}{e^{\frac{C_2}{\lambda T}}-1}\right]$ = 0

By quotient rule of differentiation,

$\frac{[e^{\frac{C_2}{\lambda T}}-1]\frac{d}{d\lambda }\left(C_1{\lambda }^{-5}\right)-\left(C_1{\lambda }^{-5}\right)\frac{d}{d\lambda }[e^{\frac{C_2}{\lambda T}}-1]}{{(e^{\frac{C_2}{\lambda T}}-1)}^2}$ = 0

$-5[e^{\frac{C_2}{\lambda T}}-1]\left(C_1{\lambda }^{-6}\right)$ – $\left(C_1{\lambda }^{-5}\right)[e^{\frac{C_2}{\lambda T}}.\frac{C_2}{T}(-\frac{1}{{\lambda }^2})]$ = 0

$-5\left(C_1{\lambda }^{-6}\right)\left[e^{\frac{C_2}{\lambda T}}\right]$ + $5\left(C_1{\lambda }^{-6}\right)$ + $\frac{C_1{C}_2{\lambda }^{-6}}{\lambda T}.e^{\frac{C_2}{\lambda T}}$ = 0

$5C_1{\lambda }^{-6}[-e^{\frac{C_2}{\lambda T}}$ + 1 + $\left(\frac{1}{5}\right)\left(\frac{C_2}{\lambda T}\right).e^{\frac{C_2}{\lambda T}}]$ = 0

$-e^{\frac{C_2}{\lambda T}}$ + 1 + $\frac{1}{5}\left(\frac{C_2}{\lambda T}\right).e^{\frac{C_2}{\lambda T}}$ = 0

Consider $\frac{C_2}{\lambda T}$ = x, thus the above equation become,

$-e^x$ + 1 + $\frac{x}{5}{e}^x$ = 0

$(\frac{x}{5}-1)e^x$ + 1 = 0

Solving the equation by trial and error method, we get

x = 4.965

∴ $\frac{C_2}{\lambda T}$ = 4.965 —–[As x = $\frac{C_2}{\lambda T}$ assumed]

At maximum value of ${\left(E_{\lambda }\right)}_b$, $\lambda ={\lambda }_{\text{max}}={\lambda }_m$

∴ $\frac{C_2}{{\lambda }_{m}T}$ = 4.965

${\lambda }_{m}T$ = $\frac{C_2}{4.965}$

Putting value of $C_2$ in above equation,

${\lambda }_{m}T$ = $\frac{0.014388}{4.965}$

${\lambda }_{m}T$ = 2.897 x 10⁻³ m.K

Significance of wien’s displacement law:

This law has following significances:-

1. The law gives the relation between the peak wavelength emitted and the temperature of the blackbody.
2. It is easy to find the approximate temperature of hotter bodies by knowing the peak wavelength emitted by the body.
3. It implies that black bodies emitting peak emissive power at lower wavelengths are hotter than black bodies emitting peak emissive power at higher wavelengths.

Application:

Following are the applications of the Wien’s displacement law:-

1] Finding the approximate temperature of the stars:

The wien’s displacement law helps to find the approximate temperature of the stars. To find the approximate temperature, it is necessary to know the peak wavelength emitted by the star.

Example:- The peak wavelength emitted by the sun is 5 x 10⁻⁷ m. Thus the temperature of the sun can be calculated as follows,

${\lambda }_{m}T$ = 2.897 x 10⁻³

(5 x 10⁻⁷) x T = 2.897 x 10⁻³

T = 5794 K

T ≈ 5800 K

Thus in this way, it is easy to find the approximate temperature of the stars situated at a too far distance.

2] Incandescent bulb:

An incandescent bulb has a filament that is heated to a higher temperature to provide a spectrum of light.

The Wien’s displacement law helps to determine the relation between the temperature of the bulb and the wavelength of the required spectrum of light.

If the temperature of the filament is lower then it produces a reddish peak wavelength and as the temperature of the filament increases, the peak wavelength moves toward the bluish region of the visible spectrum (wavelength decreases).

Wien’s displacement law solved examples:

1. Consider the earth is at a temperature of 288 K. Find the peak wavelength of energy emitted by the earth.

Given:
T = 288 K

Solution:-

By using wien’s displament law, the peak wavelength of the energy emitted from the earth is given by,

${\lambda }_{m}T$ = 2.897 x 10⁻³

${\lambda }_m$ x 288 = 2.897 x 10⁻³

${\lambda }_m$ = 1.005 x 10⁻⁵ m

Thus the earth is radiating the energy with a peak wavelength of 1.005 x 10⁻⁵ m.

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2. A star emits radiation with a peak wavelength of 10⁻⁶ m. Find the approximate temperature of the star.

Given:
${\lambda }_m$ = 10⁻⁶ m

Solution:-

Using wien’s displacement law,

${\lambda }_{m}T$ = 2.897 x 10⁻³

10⁻⁶ x T = 2.897 x 10⁻³

T = 2897 K

Thus the approximate temperature of the star is 2897 K.

FAQs:

1. What is the difference between Stefan Boltzmann law and Wien’s law?

   The Stefan Boltzmann law gives the relation between the temperature and emissive power of the black body whereas the Wien’s displacement law gives the relation between the temperature and the peak wavelength emitted by the black body.

2. What are the limitations of Wien’s displacement law?

   The wien’s displacement law gives inaccurate predictions at the higher wavelength and at the lower temperature.

3. Why is Wien’s law important?

   The wien’s law provides the relation between the temperature of the black body and the wavelength at which the black body emits the peak energy.

4. What is the dimensional formula of Wien’s constant?

   The dimensional formula of the Wien’s constant is [M¹ K¹].

We hope that this article has answered many of your questions about Wien’s displacement law. If you enjoyed this post, you should read our other articles about heat transfer.