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Zorluk: OrtaPneumatics and Gas Laws

During the operation of a pneumatic nailer, its internal firing reservoir holds a compressed air volume of 20 in320 \text{ in}^3 at an absolute pressure of 120 psi120 \text{ psi}. When the trigger is pulled, the air expands into the driver chamber at a constant temperature until the absolute pressure drops to 30 psi30 \text{ psi}. Based on Boyle's Law, what is the new total volume of the expanded air in cubic inches?

Cevap: 80 in^3

Cevap

80
According to Boyle's Law, pressure and volume are inversely proportional at a constant temperature (P1×V1=P2×V2P_1 \times V_1 = P_2 \times V_2). The pressure drops from 120 psi to 30 psi, which is a factor of 4 decrease. Because the relationship is inversely proportional, the volume must increase by a factor of 4. Therefore, the initial volume of 20 cubic inches is multiplied by 4, resulting in a final volume of 80 cubic inches.

Adım Adım Çözüm

1
Identify the initial and final states of the gas.
Initial pressure P1=120 psiP_1 = 120 \text{ psi}, Initial volume V1=20 in3V_1 = 20 \text{ in}^3. Final pressure P2=30 psiP_2 = 30 \text{ psi}.
Establishing the known parameters is required before applying the appropriate gas law.
2
Apply Boyle's Law formula.
P1×V1=P2×V2P_1 \times V_1 = P_2 \times V_2
Since the temperature is constant, Boyle's Law relates the pressure and volume of the trapped air.
3
Substitute values and solve for the unknown final volume (V2V_2).
120×20=30×V2→2400=30×V2→V2=80120 \times 20 = 30 \times V_2 \rightarrow 2400 = 30 \times V_2 \rightarrow V_2 = 80
Isolating V2V_2 provides the final expanded volume of the air.

Anahtar Kavram

Boyle's Law states that the absolute pressure of a confined gas is inversely proportional to its volume, provided the temperature remains constant.

Alternatif Yöntem

Use proportional reasoning: Observe that the final pressure (30 psi) is exactly one-quarter (1/4) of the initial pressure (120 psi). Because pressure and volume are inversely proportional under constant temperature, the final volume must be four times (4x) the initial volume. 4 × 20 = 80.
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Pneumatics and Gas Laws

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