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Zorluk: ZorPneumatics and Gas Laws

A heavy-duty pneumatic shock absorber relies on a sealed chamber of air to dampen impacts. Before an impact, the air inside the chamber has an absolute pressure of 90 psi90 \text{ psi}, occupies a volume of 60 in360 \text{ in}^3, and is at a temperature of 300 K300 \text{ K}. During a severe impact, the piston rapidly compresses the air volume to 20 in320 \text{ in}^3, causing the internal temperature to spike to 400 K400 \text{ K}. Assuming no air escapes, what is the new absolute pressure inside the shock absorber?

  1. A
    40 psi40 \text{ psi}
  2. B
    120 psi120 \text{ psi}
  3. C
    270 psi270 \text{ psi}
  4. 360 psi360 \text{ psi}Cevap

Cevap

360 psi360 \text{ psi}
The correct answer correctly applies the Combined Gas Law. The volume is reduced to one-third of its original size, which alone would triple the pressure to 270 psi270 \text{ psi}. Simultaneously, the absolute temperature increases by a factor of 4/34/3, which further scales the pressure to 270×43=360 psi270 \times \frac{4}{3} = 360 \text{ psi}.

Adım Adım Çözüm

1
Identify the initial state (P1,V1,T1P_1, V_1, T_1) and final state (V2,T2V_2, T_2) variables from the problem.
P1=90 psiP_1 = 90 \text{ psi}, V1=60 in3V_1 = 60 \text{ in}^3, T1=300 KT_1 = 300 \text{ K}; V2=20 in3V_2 = 20 \text{ in}^3, T2=400 KT_2 = 400 \text{ K}.
Establishing known parameters is required to use the Combined Gas Law.
2
Set up the Combined Gas Law equation: P1×V1T1=P2×V2T2\frac{P_1 \times V_1}{T_1} = \frac{P_2 \times V_2}{T_2}.
90×60300=P2×20400\frac{90 \times 60}{300} = \frac{P_2 \times 20}{400}
Since both volume and temperature are changing simultaneously, the combined relationship must be used.
3
Rearrange the formula to isolate the final pressure (P2P_2).
P2=P1×(V1V2)×(T2T1)P_2 = P_1 \times \left(\frac{V_1}{V_2}\right) \times \left(\frac{T_2}{T_1}\right)
This setup groups the variables into easy-to-calculate ratios.
4
Substitute the values and calculate the final result.
P2=90×(6020)×(400300)=90×3×43=360 psiP_2 = 90 \times \left(\frac{60}{20}\right) \times \left(\frac{400}{300}\right) = 90 \times 3 \times \frac{4}{3} = 360 \text{ psi}.
Mathematical evaluation yields the final absolute pressure inside the shock absorber.

Anahtar Kavram

Combined Gas Law and Compressibility
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Konular

Pneumatics and Gas Laws

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