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An ideal gas undergoes a reversible adiabatic expansion in a piston-cylinder system. The initial pressure is $\mathrm{4.0 \: atm,}$ and the initial volume is $\mathrm{40 \: liters}$ . During the expansion, the volume increases to $\mathrm{80 \: liters}$ . Calculate the final pressure of the gas.

Given the heat capacity ratio $\mathrm{(Cp/Cv)}$ for the gas is 1.25.
Option: 1
$\mathrm{3\: atm}$

Option: 2
$\mathrm{4\: atm}$

Option: 3
$\mathrm{2\: atm}$

Option: 4
$\mathrm{2.5\: atm}$

Answers (1)

best_answer

Given data:

Initial pressure $\mathrm{\left(P_i\right)=4.0 \mathrm{~atm}}$

Initial volume $\mathrm{\left(V_i\right)=40\: liters}$

Final volume $\mathrm{\left(V_f\right)=80\: liters}$

Heat capacity ratio $\mathrm{\left(C_p / C_v\right)=1.25}$

For a reversible adiabatic process, the relationship between initial and final conditions is given by:

$\mathrm{ P_i \cdot V_i^\gamma=P_f \cdot V_f^\gamma }$
Solving for the final pressure $\mathrm{ \left(P_f\right) : }$

$\mathrm{ P_f=\frac{P_i \cdot V_i^\gamma}{V_f^\gamma} }$

Substituting the values and solving:

$\mathrm{ P_f=\frac{(4.0 \mathrm{~atm}) \cdot(40 \text { liters })^{1.25}}{(80 \text { liters })^{1.25}} \approx 2.00 \mathrm{~atm} }$

Therefore, the final pressure of the gas is approximately $\mathrm{ 2.00 \mathrm{~atm} }$

So, the correct option is 3.

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