If a drop of liquid breaks into smaller droplets, it results in lowering of temperature of the droplets. Let a drop of radius R, break into N small droplets each of radius r. Estimate the drop in temperature.

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Payal Gupta | Contributor-Level 10

4 months ago

This is a short answer type question as classified in NCERT Exemplar

When a big drop of radius R, breaks into N droplets each of radius r, the volume remains constant.

Volume of big drop = N $\times$ volume of each small drop

$\frac{4}{3} π R^{3} = N \times \frac{4}{3} π r^{3}$

R³ =Nr³

N=R³/r³

Now change in surface area = 4 $π R^{3}$ -N4 $π$ r²

= 4 $π (R^{2} - N r^{2})$

Energy released = T $\times ? A$ = S $\times 4 π (R^{2} - N r^{2})$

Due to releasing of this energy , the temperature is lowered.

If $ρ$ is the density and s is specific heat of liquid and its temperature is lowered by $? θ$ , then energy released = ms $? θ$

T $\times 4 π$ ( $R^{2} - N r^{2}$ )= $(\frac{4}{3} \times R^{3} \times ρ) s ? θ$

$? θ$ = $\frac{T \times 4 π (R^{2} - N r^{2})}{\frac{4}{3} π R^{3} ρ s}$

= $\frac{3 T}{ρ s} [\frac{R^{2}}{R^{3}} - \frac{N r^{2}}{R^{3}}]$

= $\frac{3 T}{ρ s} [\frac{1}{R} - \frac{\frac{R^{3}}{r^{3}} \times r^{2}}{R^{3}}]$

$\frac{3 T}{ρ s}$ $[\frac{1}{R} - \frac{1}{r}]$

This is a short answer type question as classified in NCERT ExemplarWhen a big drop of radius R, breaks into N droplets each of radius r, the volume remains constant.Volume of big drop = N <math> <mo>×</mo> </math> volume of each small drop <math> <mfrac> <mrow> <mrow> <mn>4</mn> </mrow> </mrow> <mrow> <mrow> <mn>3</mn> </mrow> </mrow> </mfrac> <mi>π</mi> <msup> <mrow> <mrow> <mi>R</mi> </mrow> </mrow> <mrow> <mrow> <mn>3</mn> </mrow> </mrow> </msup> <mo>=</mo> <mi>N</mi> <mo>×</mo> <mfrac> <mrow> <mrow> <mn>4</mn> </mrow> </mrow> <mrow> <mrow> <mn>3</mn> </mrow> </mrow> </mfrac> <mi>π</mi> <msup> <mrow> <mrow> <mi>r</mi> </mrow> </mrow> <mrow> <mrow> <mn>3</mn> </mrow> </mrow> </msup> </math> R3 =Nr3N=R3/r3Now change in surface area = 4 <math> <mi>π</mi> <msup> <mrow> <mrow> <mi>R</mi> </mrow> </mrow> <mrow> <mrow> <mn>3</mn> </mrow> </mrow> </msup> </math> -N4 <math> <mi>π</mi> </math> r2= 4 <math> <mi>π</mi> <mfenced separators="|"> <mrow> <mrow> <msup> <mrow> <mrow> <mi>R</mi> </mrow> </mrow> <mrow> <mrow> <mn>2</mn> </mrow> </mrow> </msup> <mo>-</mo> <mi>N</mi> <msup> <mrow> <mrow> <mi>r</mi> </mrow> </mrow> <mrow> <mrow> <mn>2</mn> </mrow> </mrow> </msup> </mrow> </mrow> </mfenced> </math> Energy released = T <math> <mo>×</mo> <mo>?</mo> <mi>A</mi> </math>  = S <math> <mo>×</mo> <mn>4</mn> <mi>π</mi> <mfenced separators="|"> <mrow> <mrow> <msup> <mrow> <mrow> <mi>R</mi> </mrow> </mrow> <mrow> <mrow> <mn>2</mn> </mrow> </mrow> </msup> <mo>-</mo> <mi>N</mi> <msup> <mrow> <mrow> <mi>r</mi> </mrow> </mrow> <mrow> <mrow> <mn>2</mn> </mrow> </mrow> </msup> </mrow> </mrow> </mfenced> </math> Due to releasing of this energy , the temperature is lowered.If <math> <mi>ρ</mi> </math>  is the density and s is specific heat of liquid and its temperature is lowered by <math> <mo>?</mo> <mi>θ</mi> </math> , then energy released = ms <math> <mo>?</mo> <mi>θ</mi> </math> T <math><mo>×</mo><mn>4</mn><mi>π</mi></math> ( <math><msup><mrow><mrow><mi>R</mi></mrow></mrow><mrow><mrow><mn>2</mn></mrow></mrow></msup><mo>-</mo><mi>N</mi><msup><mrow><mrow><mi>r</mi></mrow></mrow><mrow><mrow><mn>2</mn></mrow></mrow></msup></math> )= <math><mo>(</mo><mfrac><mrow><mrow><mn>4</mn></mrow></mrow><mrow><mrow><mn>3</mn></mrow></mrow></mfrac><mo>×</mo><msup><mrow><mrow><mi>R</mi></mrow></mrow><mrow><mrow><mn>3</mn></mrow></mrow></msup><mo>×</mo><mi>ρ</mi><mo>)</mo><mi>s</mi><mo>?</mo><mi>θ</mi></math><math><mo>?</mo><mi>θ</mi></math> = <math><mfrac><mrow><mrow><mi>T</mi><mo>×</mo><mn>4</mn><mi>π</mi><mo>(</mo><msup><mrow><mrow><mi>R</mi></mrow></mrow><mrow><mrow><mn>2</mn></mrow></mrow></msup><mo>-</mo><mi>N</mi><msup><mrow><mrow><mi>r</mi></mrow></mrow><mrow><mrow><mn>2</mn></mrow></mrow></msup><mo>)</mo></mrow></mrow><mrow><mrow><mfrac><mrow><mrow><mn>4</mn></mrow></mrow><mrow><mrow><mn>3</mn></mrow></mrow></mfrac><mi>π</mi><msup><mrow><mrow><mi>R</mi></mrow></mrow><mrow><mrow><mn>3</mn></mrow></mrow></msup><mi>ρ</mi><mi>s</mi></mrow></mrow></mfrac></math>= <math><mfrac><mrow><mrow><mn>3</mn><mi>T</mi></mrow></mrow><mrow><mrow><mi>ρ</mi><mi>s</mi></mrow></mrow></mfrac><mfenced open="[" close="]" separators="|"><mrow><mrow><mfrac><mrow><mrow><msup><mrow><mrow><mi>R</mi></mrow></mrow><mrow><mrow><mn>2</mn></mrow></mrow></msup></mrow></mrow><mrow><mrow><msup><mrow><mrow><mi>R</mi></mrow></mrow><mrow><mrow><mn>3</mn></mrow></mrow></msup></mrow></mrow></mfrac><mo>-</mo><mfrac><mrow><mrow><mi>N</mi><msup><mrow><mrow><mi>r</mi></mrow></mrow><mrow><mrow><mn>2</mn></mrow></mrow></msup></mrow></mrow><mrow><mrow><msup><mrow><mrow><mi>R</mi></mrow></mrow><mrow><mrow><mn>3</mn></mrow></mrow></msup></mrow></mrow></mfrac></mrow></mrow></mfenced></math>= <math><mfrac><mrow><mrow><mn>3</mn><mi>T</mi></mrow></mrow><mrow><mrow><mi>ρ</mi><mi>s</mi></mrow></mrow></mfrac><mo>[</mo><mfrac><mrow><mrow><mn>1</mn></mrow></mrow><mrow><mrow><mi>R</mi></mrow></mrow></mfrac><mo>-</mo><mi></mi><mfrac><mrow><mrow><mfrac><mrow><mrow><msup><mrow><mrow><mi>R</mi></mrow></mrow><mrow><mrow><mn>3</mn></mrow></mrow></msup></mrow></mrow><mrow><mrow><msup><mrow><mrow><mi>r</mi></mrow></mrow><mrow><mrow><mn>3</mn></mrow></mrow></msup></mrow></mrow></mfrac><mo>×</mo><msup><mrow><mrow><mi>r</mi></mrow></mrow><mrow><mrow><mn>2</mn></mrow></mrow></msup></mrow></mrow><mrow><mrow><msup><mrow><mrow><mi>R</mi></mrow></mrow><mrow><mrow><mn>3</mn></mrow></mrow></msup></mrow></mrow></mfrac><mo>]</mo></math><math><mfrac><mrow><mrow><mn>3</mn><mi>T</mi></mrow></mrow><mrow><mrow><mi>ρ</mi><mi>s</mi></mrow></mrow></mfrac></math><math><mfenced open="[" close="]" separators="|"><mrow><mrow><mfrac><mrow><mrow><mn>1</mn></mrow></mrow><mrow><mrow><mi>R</mi></mrow></mrow></mfrac><mo>-</mo><mfrac><mrow><mrow><mn>1</mn></mrow></mrow><mrow><mrow><mi>r</mi></mrow></mrow></mfrac></mrow></mrow></mfenced></math>

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If a drop of liquid breaks into smaller droplets, it results in lowering of temperature of the droplets. Let a drop of radius R, break into N small droplets each of radius r. Estimate the drop in temperature.

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