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Chapter 1: Problem 98

Air is enclosed by a rigid cylinder containing a piston. \(A\) pressure gage attached to the cylinder indicates an initial reading of 25 psi. Determine the reading on the gage when the piston has compressed the air to one-third its original volume. Assume the compression process to be isothemal and the local atmospheric pressure to be 14.7 psi.

Short Answer

Expert verified
The reading on the gage when the piston has compressed the air to one-third its original volume will be 104.4 psi.

Step by step solution

01

Convert Initial Pressure to Absolute Pressure

The pressure gage gives the pressure of the air relative to the atmospheric pressure. To find the absolute pressure, add the gage reading to the local atmospheric pressure: \[P_1 = 25 \, \text{psi} + 14.7 \, \text{psi} = 39.7 \, \text{psi}\]
02

Apply Ideal Gas Law

Given the process is isothermal (constant temperature), the ideal gas law states that the product of pressure and volume is a constant. Thus, if the volume is decreased to one-third, the pressure must increase three times to maintain the constant. We have \(P_1V_1 = P_2V_2\), which implies \(P_2 = P_1 \cdot V_1 / V_2\). Since \(V_1 / V_2 = 3\), we have \(P_2 = P_1 \cdot 3\). So, \(P_2 = 39.7 \, \text{psi} \cdot 3 = 119.1 \, \text{psi}\).
03

Convert Absolute Pressure to Gage Pressure

Now you subtract the atmospheric pressure from the absolute pressure to get the gage pressure: \[P_{2,\text{gage}} = P_2 - P_{\text{atm}} = 119.1 \, \text{psi} - 14.7 \, \text{psi} = 104.4 \, \text{psi}\].

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