Consider that an ideal gas (n moles) is expanding in a process given by P = f ( V ), which passes through a point (Po, V0 ). Show that the gas is absorbing heat at (P0, Vo ) if the slope of the curve P = f (V ) is larger than the slope of the adiabat passing through (Po, V0 ).

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Thermodynamics physics ncert solutions class 11th Physics Class 11th Physics NCERT Exemplar Solutions Class 11th Chapter Twelve

Consider that an ideal gas (n moles) is expanding in a process given by P = f ( V ), which passes through a point (P_o, V₀ ). Show that the gas is absorbing heat at (P₀, V_o ) if the slope of the curve P = f (V ) is larger than the slope of the adiabat passing through (P_o, V₀ ).

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Payal Gupta | Contributor-Level 10

4 months ago

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

Slope of the curve = f(V) , where V is the volume

Slope of P = f(V) curve at ((P_o, V₀ )= f(V_o)

Slope of adiabatic at (P_o, V₀ )= k(-Y)V_o^-1-Y =-YP_o/V_o

Now heat absorbed in the process P= f(V)

dQ=dU+dW= nC_vdT+pdV

pV=nRT

T= pV/nR

T= $\frac{1}{n R} [f (V) + V f^{'} (V) d V]$

$\frac{d Q}{d V} =$ nC_{v
$\frac{d T}{d V} + p \frac{d V}{d V} = n C_{V} \frac{d T}{d V} + P$}

$\frac{C_{V}}{R} [f (V_{o} + V_{0} f^{'} (V) + f (V)]$

After solving we get

${[\frac{d Q}{d V}]}_{v = v o} =$ $[\frac{1}{Y - 1} + 1] f (V_{o}) + \frac{V_{o} f^{' V_{o}}}{Y - 1}$

= $\frac{Y}{Y - 1} P_{o} + \frac{V_{o}}{Y - 1} f^{'} (V_{o})$

Heat is absorbed where dQ/dV>0 when gas expands

Hence YP_o+V_of’(V_o)>0 or f’(V_o)>(-Y $\frac{P_{o}}{V_{o}}$ )

This is a long answer type question as classified in NCERT ExemplarSlope of the curve = f(V) , where V is the volumeSlope of P = f(V) curve at ((Po, V0 )= f(Vo)Slope of adiabatic at (Po, V0 )= k(-Y)Vo-1-Y =-YPo/VoNow heat absorbed in the process P= f(V)dQ=dU+dW= nCvdT+pdVpV=nRTT= pV/nRT= <math> <mfrac> <mrow> <mrow> <mn>1</mn> </mrow> </mrow> <mrow> <mrow> <mi>n</mi> <mi>R</mi> </mrow> </mrow> </mfrac> <mfenced open="[" close="]" separators="|"> <mrow> <mrow> <mi>f</mi> <mfenced separators="|"> <mrow> <mrow> <mi>V</mi> </mrow> </mrow> </mfenced> <mo>+</mo> <mi>V</mi> <msup> <mrow> <mrow> <mi>f</mi> </mrow> </mrow> <mrow> <mrow> <mi>'</mi> </mrow> </mrow> </msup> <mfenced separators="|"> <mrow> <mrow> <mi>V</mi> </mrow> </mrow> </mfenced> <mi>d</mi> <mi>V</mi> </mrow> </mrow> </mfenced> </math> <math> <mfrac> <mrow> <mrow> <mi>d</mi> <mi>Q</mi> </mrow> </mrow> <mrow> <mrow> <mi>d</mi> <mi>V</mi> </mrow> </mrow> </mfrac> <mo>=</mo> </math> nCv <math> <mfrac> <mrow> <mrow> <mi>d</mi> <mi>T</mi> </mrow> </mrow> <mrow> <mrow> <mi>d</mi> <mi>V</mi> </mrow> </mrow> </mfrac> <mo>+</mo> <mi>p</mi> <mfrac> <mrow> <mrow> <mi>d</mi> <mi>V</mi> </mrow> </mrow> <mrow> <mrow> <mi>d</mi> <mi>V</mi> </mrow> </mrow> </mfrac> <mo>=</mo> <mi>n</mi> <msub> <mrow> <mrow> <mi>C</mi> </mrow> </mrow> <mrow> <mrow> <mi>V</mi> </mrow> </mrow> </msub> <mfrac> <mrow> <mrow> <mi>d</mi> <mi>T</mi> </mrow> </mrow> <mrow> <mrow> <mi>d</mi> <mi>V</mi> </mrow> </mrow> </mfrac> <mo>+</mo> <mi>P</mi> </math> <math> <mfrac> <mrow> <mrow> <msub> <mrow> <mrow> <mi>C</mi> </mrow> </mrow> <mrow> <mrow> <mi>V</mi> </mrow> </mrow> </msub> </mrow> </mrow> <mrow> <mrow> <mi>R</mi> </mrow> </mrow> </mfrac> <mo>[</mo> <mi>f</mi> <mo>(</mo> <msub> <mrow> <mrow> <mi>V</mi> </mrow> </mrow> <mrow> <mrow> <mi>o</mi> </mrow> </mrow> </msub> <mo>+</mo> <msub> <mrow> <mrow> <mi>V</mi> </mrow> </mrow> <mrow> <mrow> <mn>0</mn> </mrow> </mrow> </msub> <msup> <mrow> <mrow> <mi>f</mi> </mrow> </mrow> <mrow> <mrow> <mi>'</mi> </mrow> </mrow> </msup> <mfenced separators="|"> <mrow> <mrow> <mi>V</mi> </mrow> </mrow> </mfenced> <mo>+</mo> <mi>f</mi> <mo>(</mo> <mi>V</mi> <mo>)</mo> <mo>]</mo> </math> After solving we get <math> <msub> <mrow> <mrow> <mfenced open="[" close="]" separators="|"> <mrow> <mrow> <mfrac> <mrow> <mrow> <mi>d</mi> <mi>Q</mi> </mrow> </mrow> <mrow> <mrow> <mi>d</mi> <mi>V</mi> </mrow> </mrow> </mfrac> </mrow> </mrow> </mfenced> </mrow> </mrow> <mrow> <mrow> <mi>v</mi> <mo>=</mo> <mi>v</mi> <mi>o</mi> </mrow> </mrow> </msub> <mo>=</mo> </math> <math> <mfenced open="[" close="]" separators="|"> <mrow> <mrow> <mfrac> <mrow> <mrow> <mn>1</mn> </mrow> </mrow> <mrow> <mrow> <mi>Y</mi> <mo>-</mo> <mn>1</mn> </mrow> </mrow> </mfrac> <mo>+</mo> <mn>1</mn> </mrow> </mrow> </mfenced> <mi>f</mi> <mfenced separators="|"> <mrow> <mrow> <msub> <mrow> <mrow> <mi>V</mi> </mrow> </mrow> <mrow> <mrow> <mi>o</mi> </mrow> </mrow> </msub> </mrow> </mrow> </mfenced> <mo>+</mo> <mfrac> <mrow> <mrow> <msub> <mrow> <mrow> <mi>V</mi> </mrow> </mrow> <mrow> <mrow> <mi>o</mi> </mrow> </mrow> </msub> <msup> <mrow> <mrow> <mi>f</mi> </mrow> </mrow> <mrow> <mrow> <mi>'</mi> <msub> <mrow> <mrow> <mi>V</mi> </mrow> </mrow> <mrow> <mrow> <mi>o</mi> </mrow> </mrow> </msub> </mrow> </mrow> </msup> </mrow> </mrow> <mrow> <mrow> <mi>Y</mi> <mo>-</mo> <mn>1</mn> </mrow> </mrow> </mfrac> </math> = <math> <mfrac> <mrow> <mrow> <mi>Y</mi> </mrow> </mrow> <mrow> <mrow> <mi>Y</mi> <mo>-</mo> <mn>1</mn> </mrow> </mrow> </mfrac> <msub> <mrow> <mrow> <mi>P</mi> </mrow> </mrow> <mrow> <mrow> <mi>o</mi> </mrow> </mrow> </msub> <mo>+</mo> <mfrac> <mrow> <mrow> <msub> <mrow> <mrow> <mi>V</mi> </mrow> </mrow> <mrow> <mrow> <mi>o</mi> </mrow> </mrow> </msub> </mrow> </mrow> <mrow> <mrow> <mi>Y</mi> <mo>-</mo> <mn>1</mn> </mrow> </mrow> </mfrac> <msup> <mrow> <mrow> <mi>f</mi> </mrow> </mrow> <mrow> <mrow> <mi>'</mi> </mrow> </mrow> </msup> <mfenced separators="|"> <mrow> <mrow> <msub> <mrow> <mrow> <mi>V</mi> </mrow> </mrow> <mrow> <mrow> <mi>o</mi> </mrow> </mrow> </msub> </mrow> </mrow> </mfenced> </math> Heat is absorbed where dQ/dV>0  when gas expandsHence YPo+Vof’(Vo)>0  or f’(Vo)>(-Y <math> <mfrac> <mrow> <mrow> <msub> <mrow> <mrow> <mi>P</mi> </mrow> </mrow> <mrow> <mrow> <mi>o</mi> </mrow> </mrow> </msub> </mrow> </mrow> <mrow> <mrow> <msub> <mrow> <mrow> <mi>V</mi> </mrow> </mrow> <mrow> <mrow> <mi>o</mi> </mrow> </mrow> </msub> </mrow> </mrow> </mfrac> </math> )

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Consider that an ideal gas (n moles) is expanding in a process given by P = f ( V ), which passes through a point (P_o, V₀ ). Show that the gas is absorbing heat at (P₀, V_o ) if the slope of the curve P = f (V ) is larger than the slope of the adiabat passing through (P_o, V₀ ).

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