A 6.0-cm-diameter, 10-cm-long cylinder contains 100 mg of oxygen (O₂) at a pressure less than 1 atm. 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 100 kPa, it takes a 184 N force to pull the cap off. What is the temperature of the gas?

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Hello, fellow physicists today, we're gonna solve the following practice problem together. So first off, let's read the problem and highlight all the key pieces of information that we need to use in order to solve this problem. Nitrogen storage tanks are cylindrical pressure vessels used to prevent product spoilage. A cylindrical pressure vessel containing 6.22 g of nitrogen gas is sealed with a lid to prevent leaks. The pressure inside the vessel is below 1.00 multiplied by to the power of five pascals. Thus, the lid is secured by the force exerted by the atmospheric pressure. The vessel is 60 centimeters long and has a 7.00 centimeter radius. If the force needed to pull the lid is equal to 400 newtons at a location where the atmospheric pressure is 1.10 multiplied by 10 of the power of five pascals find the gas temperature. Ok. So our end goal is to find the gas temperature for nitrogen. Ok. So we're given some multiple choice answers. They're all given in the same units of degrees Celsius. Let's read them off to see what our final answer might be A is 45.0. B is 85.5 C is 148 and D is 185. So first off, let us assume that nitrogen is an ideal gas. So we can then recall and use the ideal gas Law equation, let's call it equation one. And that states that the pressure multiplied by the volume is equal to the number of moles multiplied by the universal gas constant multiplied by the temperature. So in order to find the temperature, we first need to find the number of moles because we're not given that in the problem itself. So let's start with that. So to solve for the numbers of moles, we need to recall how to find the number of moles. And we need to recall the equation which states that the number of moles is equal to the mass of the sample divided by the molar mass of the sample. So let's make a side note here that the molar mass of nitrogen. So capital M subscript N, capital N subscript two. So the molar mass of nitrogen is equal to 28 point g per mole. OK. So you can either look that up really quickly or you can calculate it on your own. But that's just the molar mass of nitrogen. So now we can plug in our known variables to solve for the number of moles. So we're given 6.22 g of, of nitrogen gas in the problem divided by the molar mass, which was 28.0 g per mole, the grams cancel out just leaving us with the moles. So then we have we put into a calculator, 0. moles. OK. Now, we must consider the area of the lid. So let's note that the area of the lid is the is a equal to. So the area is equal to pi multiplied by the radius of the vessel squared. So let's plug in our known variables to sol for the area. So area equals pi multiplied by the radius which the radius is given to us in the prom as 7.00 centimeters. But we need to convert that to meters. So in order to do that, we all, all we have to do is just take 7.00 multiplied by 10 to the power of negative two meters squared. And when we plug that into a calculator, we should get 1. multiplied by 10 to the power of negative two meters squared. Ok. So now we need to consider the volume of the vessel. So let's note that the volume of the vessel. So V the V for the volume is equal to the area multiplied by the length of the vessel. So let's plug in our known values to solve for the volume. We know the area because we just solved for it it's one point multiplied by 10 to the power of negative two meters squared, multiplied by the length which the length is given to us in the prom is centimeters. But we need to convert that to meters. So all we have to do is take 60 multiplied by 10 to the power of negative 2 m. So when we plug that into a calculator, we should get 9.24 multiplied by 10. The power of negative three meters. Cute. Ok. Let's take a breath for a second here. Ok. So now at this stage, we need to note that the pressure inside the cylinder can be calculated by subtracting the pressure created by the 400 Newton force required to remove the lid from atmospheric pressure. The pressure needed to pull the lid off or to needed to pull off the lid is given by the force divided by the area. So considering this, we can write the following, we can state that the pressure is equal to the atmospheric pressure, which is 1.10 multiplied by 10 to the power of five pascals subtracted by 400 newtons divided by the area means our force is 400 newtons. And then the area was 1.54 multiplied by 10 to the power of negative 2 m squared is equal to when you plug into a calculator, 0.840 multiplied by 10 to the fifth power pass scales. So at this stage, we're finally at the, at the end here, now we can take equation one and rearrange it. So equation one was the ideal gas law. So we need to rearrange that to solve for temperature. So when we rearrange the ideal gas law to solve for temperature, we get that the temperature is equal to the pressure multiplied by the volume divided by the number of moles multiplied by the universal gas constant. OK. So let's plug in our known variables to solve for the temperature. So we know that the pressure, yes, we just found it is 0.840 multiplied by 10 to the fifth. Power past scales multiplied by the volume which we determine to be 9.24 multiplied by 10. The power of negative 3 m cubed. We know that the number of moles is 0. males. And the numerical value for the universal gas constant is 8.31 jewels per mole multiplied by Calvin. OK. So when we plug that into a calculator, we should get 421 Kelvin. But we need to convert Kelvin to degrees Celsius because all of our multiple choice answers are in degrees Celsius. So all we have to do is take 421 and not add. But we need to subtract which gives us when we subtract that we get 148 degrees Celsius, which is our final answer. We did it, we found the temperature of the gas. So that means that our final answer has to be the letter C 148 degrees Celsius. Thank you so much for watching. Hopefully that helped and I can't wait to see you in the next video. Bye.

Verified Solution

Key Concepts

Ideal Gas Law

Pressure and Force Relationship

Volume of a Cylinder