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  • A gas in a piston-cylinder system undergoes an adiabatic expansion. The initial pressure of the gas is 8 atm and the initial volume is  <img alt="5 \mathrm{~L}" src="https://entrancecorner.onc

A gas in a piston-cylinder system undergoes an adiabatic expansion. The initial pressure of the gas is 8 atm and the initial volume is  5 \mathrm{~L} During the expansion, the volume of the gas triples. Calculate the final pressure of the gas after the expansion and the work done by the gas.

Given:
      \begin{aligned} \text { Initial pressure }\left(P_{i}\right) & =8 \mathrm{~atm} \\ \text { Initial volume }\left(V_{i}\right) & =5 \mathrm{~L} \\ \text { Final volume }\left(V_{f}\right) & =15 \mathrm{~L} \end{aligned}

   Ratio of specific heat capacities ()=1.3 for the gas

Option: 1

-24.14 L atm


Option: 2

-26.54 L atm


Option: 3

-22.24 L atm


Option: 4

-29.21 L atm


Answers (1)

best_answer

The adiabatic process for an ideal gas can be described using the relationship between pressure \mathrm{(P)}, volume \mathrm{(V)}, and the ratio of specific heat capacities () :

\mathrm{P V^{\gamma}=\text { constant }}

Step 1: Use the initial conditions to find the constant of the process:

Substitute the given values:
\mathrm{P_{i} V_{i}^{\gamma}=P_{f} V_{f}^{\gamma} }
\mathrm{P_{f}=P_{i}\left(\frac{V_{i}}{V_{f}}\right)^{\gamma}}

\mathrm{P_{f}=(8 \mathrm{~atm})\left(\frac{5 \mathrm{~L}}{15 \mathrm{~L}}\right)^{1.3}}
\mathrm{P_{f} \approx 2.153 \mathrm{~atm}}


Step 2: Calculate the work done by the gas using the formula for adiabatic expansion work:
\mathrm{W=-\frac{P_{f} V_{f}-P_{i} V_{i}}{1-}}
Substitute the given values:
\mathrm{W=-\frac{(2.153 \mathrm{~atm})(15 \mathrm{~L})-(8 \mathrm{~atm})(5 \mathrm{~L})}{1-1.3} }
\mathrm{W \approx-26.54 \mathrm{~L} \text { atm } }

nswer: After the adiabatic expansion, the final pressure of the gas is approximately  \mathrm{2.153 \mathrm{~atm}}, and the work done by the gas is approximately -26.54 \mathrm{\mathrm{L}} atm.

Posted by

HARSH KANKARIA

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