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Physics Ncert Solutions Class 12th Physics Class 12th Thermodynamics

A Carnot engine has efficiency of 50%. If the temperature of sink is reduced by 40°C, its efficiency increases by 30%. The temperature of the source will be:

Option 1 -

166.7 K

Option 2 -

255.1 K

Option 3 -

266.7 K

Option 4 -

367.7 K

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Answered by

Vishal Baghel | Contributor-Level 10

2 months ago

Correct Option - 3

Detailed Solution:

Efficiency $η = 50 %$

$\Rightarrow η = (1 - \frac{T_{2}}{T_{1}}) \times 100 = 50$

$\Rightarrow 1 - \frac{T_{2}}{T_{1}} = \frac{50}{100} \Rightarrow 1 - \frac{T_{2}}{T_{1}} = \frac{1}{2}$ -(i)

After the change :- $1 - \frac{T_{2}^{1}}{T_{1}} = \frac{50 \times (1.3)}{100} = \frac{65}{100} = . 65$ $\frac{_{}^{}}{_{}} \frac{}{} \frac{}{}$ -(ii)

$\Rightarrow 1 - (\frac{T_{2} - 40}{T_{1}}) = . 65$ --(ii)

$\Rightarrow \frac{40}{T_{1}} = . 65 - . 50 = . 15 \Rightarrow T_{1} = \frac{4000}{15} = 266.7 k$

Efficiency <math> <mrow> <mi>η</mi> <mo>=</mo> <mn>5</mn> <mn>0</mn> <mn>%</mn> </mrow> </math> <math><mrow></mrow></math> <math> <mrow> <mo>⇒</mo> <mi>η</mi> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>−</mo> <mfrac> <mrow> <msub> <mrow> <mi>T</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msub> </mrow> <mrow> <msub> <mrow> <mi>T</mi> </mrow> <mrow> <mn>1</mn> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>×</mo> <mn>1</mn> <mn>0</mn> <mn>0</mn> <mo>=</mo> <mn>5</mn> <mn>0</mn> </mrow> </math> <img ><math><mrow><mfrac><mrow><msub><mrow></mrow></msub></mrow></mfrac></mrow></math> <math> <mrow> <mo>⇒</mo> <mn>1</mn> <mo>−</mo> <mfrac> <mrow> <msub> <mrow> <mi>T</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msub> </mrow> <mrow> <msub> <mrow> <mi>T</mi> </mrow> <mrow> <mn>1</mn> </mrow> </msub> </mrow> </mfrac> <mo>=</mo> <mfrac> <mrow> <mn>5</mn> <mn>0</mn> </mrow> <mrow> <mn>1</mn> <mn>0</mn> <mn>0</mn> </mrow> </mfrac> <mo>⇒</mo> <mn>1</mn> <mo>−</mo> <mfrac> <mrow> <msub> <mrow> <mi>T</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msub> </mrow> <mrow> <msub> <mrow> <mi>T</mi> </mrow> <mrow> <mn>1</mn> </mrow> </msub> </mrow> </mfrac> <mo>=</mo> <mfrac> <mrow> <mn>1</mn> </mrow> <mrow> <mn>2</mn> </mrow> </mfrac> </mrow> </math>  -(i)After the change :- <math> <mrow> <mn>1</mn> <mo>−</mo> <mfrac> <mrow> <msubsup> <mrow> <mi>T</mi> </mrow> <mrow> <mn>2</mn> </mrow> <mrow> <mn>1</mn> </mrow> </msubsup> </mrow> <mrow> <msub> <mrow> <mi>T</mi> </mrow> <mrow> <mn>1</mn> </mrow> </msub> </mrow> </mfrac> <mo>=</mo> <mfrac> <mrow> <mn>5</mn> <mn>0</mn> <mo>×</mo> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>.</mo> <mn>3</mn> </mrow> <mo>)</mo> </mrow> </mrow> <mrow> <mn>1</mn> <mn>0</mn> <mn>0</mn> </mrow> </mfrac> <mo>=</mo> <mfrac> <mrow> <mn>6</mn> <mn>5</mn> </mrow> <mrow> <mn>1</mn> <mn>0</mn> <mn>0</mn> </mrow> </mfrac> <mo>=</mo> <mo>.</mo> <mn>6</mn> <mn>5</mn> </mrow> </math> <math><mrow><mfrac><mrow><msubsup><mrow></mrow><mrow></mrow><mrow></mrow></msubsup></mrow><mrow><msub><mrow></mrow><mrow></mrow></msub></mrow></mfrac><mfrac><mrow><mrow><mrow></mrow></mrow></mrow><mrow></mrow></mfrac><mfrac><mrow></mrow><mrow></mrow></mfrac></mrow></math> -(ii) <math> <mrow> <mo>⇒</mo> <mn>1</mn> <mo>−</mo> <mrow> <mo>(</mo> <mrow> <mfrac> <mrow> <msub> <mrow> <mi>T</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msub> <mo>−</mo> <mn>4</mn> <mn>0</mn> </mrow> <mrow> <msub> <mrow> <mi>T</mi> </mrow> <mrow> <mn>1</mn> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mo>.</mo> <mn>6</mn> <mn>5</mn> </mrow> </math> --(ii) <math> <mrow> <mo>⇒</mo> <mfrac> <mrow> <mn>4</mn> <mn>0</mn> </mrow> <mrow> <msub> <mrow> <mi>T</mi> </mrow> <mrow> <mn>1</mn> </mrow> </msub> </mrow> </mfrac> <mo>=</mo> <mo>.</mo> <mn>6</mn> <mn>5</mn> <mo>−</mo> <mo>.</mo> <mn>5</mn> <mn>0</mn> <mo>=</mo> <mo>.</mo> <mn>1</mn> <mn>5</mn> <mo>⇒</mo> <msub> <mrow> <mi>T</mi> </mrow> <mrow> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mfrac> <mrow> <mn>4</mn> <mn>0</mn> <mn>0</mn> <mn>0</mn> </mrow> <mrow> <mn>1</mn> <mn>5</mn> </mrow> </mfrac> <mo>=</mo> <mn>2</mn> <mn>6</mn> <mn>6</mn> <mo>.</mo> <mn>7</mn> <mi>k</mi> </mrow> </math>

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...more

T₁ = 227°C

= 500k

T₂ =?

$\frac{Q_{1}}{T_{1}} = \frac{Q_{2}}{T_{2}}$

$\frac{300}{500} = \frac{225}{T_{2}}$

$T_{2} = \frac{500 * 225}{300}$

= 5 * 75

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T₂ = 102°C

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A Carnot engine has efficiency of 50%. If the temperature of sink is reduced by 40°C, its efficiency increases by 30%. The temperature of the source will be:

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