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A 6.0cm-diameter,10cm-longcylinder contains 100mgof oxygen (O2) at a pressure less than 1atm. The cap on one end of the cylinder is held in place only by the pressure of the air. One day when the atmospheric pressure is 100kPa, it takes a 184Nforce to pull the cap off. What is the temperature of the gas?

Short Answer

Expert verified

The temperature of the gas is380K

Step by step solution

01

Defining Ideal Gas law

Given : A 6.0-cm-diameter, 10-cm-long cylinder contains 100mgof oxygen at a pressure less than 1atm. When atmospheric pressure is 100kPa, it takes a 184Nforce to pull the cap off.

The ideal gas law (PV=nRT) relates the macroscopic properties of ideal gases.

Since the pressurep comes from inside the gas, we can apply the ideal gas law to find an expression for the temperature
PV=nRTT=PVnR
Knowing that we are giving the mass and not the number of moles, we can rewrite the previous formula as follows.
T=pMVmR, where Mis the molar mass of the gas inside and mis the mass of that gas.

02

Finding the temperature of the gas 

Based onNewton's first law and the definition of pressure, we can write for the cap that
pA+F=paAp=pa-FA
Where pis the internal pressure, pais the atmospheric pressure,

A=πR²=πD²4is the area of the cap andFis the force we have to apply to move the cap.
We can now substitute the expression we found for pressure into our result for temperature:

T=pMVmR=(pa-FA)MVmR

Remembering that for a cylinderV=Ah, we can further simplify to If we apply our numerical input data, we find the value for the temperature inside as follows.
T=(pa-FA)MVmR=(pa-FA)MAhmR=(Apa-F)MhmRT=(πD²4pa-F)MhmR=Mh(πD²pa-4F)4mR

Inserting the values, we getT=380k

T=380K

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