12.4 A cylinder with a movable piston contains 3 moles of hydrogen at standard temperature and pressure. The walls of the cylinder are made of a heat insulator, and the piston is insulated by having a pile of sand on it. By what factor does the pressure of the gas increase if the gas is compressed to half its original volume ?

<p>The cylinder is completely insulated from its surroundings. As a result, no heat is exchanged between the system &#40;cylinder&#41; and its surroundings. Thus the process is ‘Adiabatic’.</p><p>Let the initial and final pressure inside the cylinder be&nbsp;<span title="Click to copy mathml"><math><msub><mrow><mrow><mi>P</mi></mrow></mrow><mrow><mrow><mn>1</mn><mi></mi></mrow></mrow></msub></math></span>&nbsp;&amp;&nbsp;<span title="Click to copy mathml"><math><msub><mrow><mrow><mi>P</mi></mrow></mrow><mrow><mrow><mn>2</mn></mrow></mrow></msub></math></span>&nbsp;and volume be&nbsp;<span title="Click to copy mathml"><math><msub><mrow><mrow><mi>V</mi></mrow></mrow><mrow><mrow><mn>1</mn><mi></mi></mrow></mrow></msub></math></span>&nbsp;&amp;&nbsp;<span title="Click to copy mathml"><math><msub><mrow><mrow><mi>V</mi></mrow></mrow><mrow><mrow><mn>2</mn></mrow></mrow></msub></math></span>&nbsp;.</p><p>Ratio of specific heat,&nbsp;<span title="Click to copy mathml"><math><mi>γ</mi></math></span>&nbsp;= 1.4</p><p>For an adiabatic process, we know&nbsp;<span title="Click to copy mathml"><math><msub><mrow><mrow><mi>P</mi></mrow></mrow><mrow><mrow><mn>1</mn><mi></mi></mrow></mrow></msub><msubsup><mrow><mrow><mi>V</mi></mrow></mrow><mrow><mrow><mn>1</mn></mrow></mrow><mrow><mrow><mi>γ</mi></mrow></mrow></msubsup></math></span>&nbsp;=&nbsp;<span title="Click to copy mathml"><math><msub><mrow><mrow><mi>P</mi></mrow></mrow><mrow><mrow><mn>2</mn><mi></mi></mrow></mrow></msub><msubsup><mrow><mrow><mi>V</mi></mrow></mrow><mrow><mrow><mn>2</mn></mrow></mrow><mrow><mrow><mi>γ</mi></mrow></mrow></msubsup></math></span></p><p>It is given&nbsp;<span title="Click to copy mathml"><math><msub><mrow><mrow><mi>V</mi></mrow></mrow><mrow><mrow><mn>2</mn></mrow></mrow></msub><mo>=</mo></math></span>&nbsp;<span title="Click to copy mathml"><math><mfrac><mrow><mrow><msub><mrow><mrow><mi>V</mi></mrow></mrow><mrow><mrow><mn>1</mn><mi></mi></mrow></mrow></msub></mrow></mrow><mrow><mrow><mn>2</mn></mrow></mrow></mfrac></math></span></p><p>Hence&nbsp;<span title="Click to copy mathml"><math><msub><mrow><mrow><mi>P</mi></mrow></mrow><mrow><mrow><mn>1</mn><mi></mi></mrow></mrow></msub><msubsup><mrow><mrow><mi>V</mi></mrow></mrow><mrow><mrow><mn>1</mn></mrow></mrow><mrow><mrow><mi>γ</mi></mrow></mrow></msubsup></math></span>&nbsp;=&nbsp;<span title="Click to copy mathml"><math><msub><mrow><mrow><mi>P</mi></mrow></mrow><mrow><mrow><mn>2</mn><mi></mi></mrow></mrow></msub><msup><mrow><mrow><mfrac><mrow><mrow><msub><mrow><mrow><mi>V</mi></mrow></mrow><mrow><mrow><mn>1</mn><mi></mi></mrow></mrow></msub></mrow></mrow><mrow><mrow><mn>2</mn></mrow></mrow></mfrac></mrow></mrow><mrow><mrow><mi>γ</mi></mrow></mrow></msup></math></span>&nbsp;or&nbsp;<span title="Click to copy mathml"><math><mfrac><mrow><mrow><msub><mrow><mrow><mi>P</mi></mrow></mrow><mrow><mrow><mn>2</mn></mrow></mrow></msub></mrow></mrow><mrow><mrow><msub><mrow><mrow><mi>P</mi></mrow></mrow><mrow><mrow><mn>1</mn><mi></mi></mrow></mrow></msub></mrow></mrow></mfrac></math></span>&nbsp;= (&nbsp;<span title="Click to copy mathml"><math><msubsup><mrow><mrow><mi>V</mi></mrow></mrow><mrow><mrow><mn>1</mn></mrow></mrow><mrow><mrow><mi>γ</mi></mrow></mrow></msubsup><mo>)</mo><mo>/</mo></math></span>&nbsp;(&nbsp;<span title="Click to copy mathml"><math><mi></mi><msup><mrow><mrow><mfrac><mrow><mrow><msub><mrow><mrow><mi>V</mi></mrow></mrow><mrow><mrow><mn>1</mn><mi></mi></mrow></mrow></msub></mrow></mrow><mrow><mrow><mn>2</mn></mrow></mrow></mfrac></mrow></mrow><mrow><mrow><mi>γ</mi></mrow></mrow></msup><mo>)</mo></math></span>&nbsp;=&nbsp;<span title="Click to copy mathml"><math><msup><mrow><mrow><mn>2</mn></mrow></mrow><mrow><mrow><mi>γ</mi></mrow></mrow></msup></math></span>&nbsp;=&nbsp;<span title="Click to copy mathml"><math><msup><mrow><mrow><mn>2</mn></mrow></mrow><mrow><mrow><mn>1.4</mn></mrow></mrow></msup></math></span>&nbsp;= 2.639</p><p>Hence the pressure increases by a factor of 2.639</p>

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