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Chapter 18: Problem 62

A vertical cylindrical tank contains 1.80 \(\mathrm{mol}\) of an ideal gas under a pressure of 1.00 atm at \(20.0^{\circ} \mathrm{C}\) . The round part of the tank has a radius of \(10.0 \mathrm{cm},\) and the gas is supporting a piston that can move up and down in the cylinder without friction. (a) What is the mass of this piston? (b) How tall is the column of gas that is supporting the piston?

Short Answer

Expert verified
The mass of the piston is 324.41 kg. The height of the column of gas is 1.37 meters.

Step by step solution

01

Understand the Physical System

The system consists of a cylindrical tank with a movable piston supported by an ideal gas inside it. We need to determine the mass of the piston based on the pressure exerted by the gas, as well as the height of the column of gas.
02

Calculate the Force Exerted by the Gas

The force exerted by the gas on the piston can be calculated using the formula \( F = P \times A \), where \( P \) is the pressure of the gas, and \( A \) is the area of the piston. The pressure \( P \) is 1.00 atm, and we need to convert this to Pascals: 1.00 atm = 101325 Pa. Area \( A \) can be calculated as the area of a circle, \( A = \pi r^2 \), where \( r = 10.0 \) cm = 0.10 m. So, \( A = \pi \times (0.10)^2 \).
03

Convert Area and Force Calculation

Calculate the area of the piston: \[ A = \pi \times (0.10)^2 = 0.0314 \text{ m}^2. \] Using the pressure, compute the force: \[ F = 101325 \times 0.0314 = 3183 \text{ N}. \] This is the force exerted by the gas on the piston.
04

Determine the Mass of the Piston

The force exerted by the gas is equal to the weight of the piston, which is the force due to gravity acting on it. This can be expressed as \( F = m \times g \), where \( m \) is the mass of the piston and \( g = 9.81 \text{ m/s}^2 \). Solve for \( m \): \[ m = \frac{F}{g} = \frac{3183}{9.81} = 324.41 \text{ kg}. \]
05

Use Ideal Gas Law to Find Volume

To find the height of the gas column, we first find the volume of the gas using the ideal gas law \( PV = nRT \). Here, \( P = 101325 \) Pa, \( n = 1.80 \) mol, \( R = 8.314 \text{ J/(mol K)} \) and \( T = 20.0^{\circ} \text{C} = 293 \text{ K}. \)\[ V = \frac{nRT}{P} = \frac{1.80 \times 8.314 \times 293}{101325} \approx 0.043 \text{ m}^3. \]
06

Calculate Height of Gas Column

With the volume of the cylinder, calculate the height of the column of gas using \( V = A \times h \). We already calculated \( A = 0.0314 \text{ m}^2 \). Solve for \( h \): \[ h = \frac{V}{A} = \frac{0.043}{0.0314} \approx 1.37 \text{ m}. \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Cylindrical Tank Dynamics
A cylindrical tank is an essential component in various engineering applications, including gas storage. It's crucial to understand how it operates, especially when it involves a movable piston. This setup allows the gas inside the tank to expand or contract freely with the piston adjusting its position accordingly. The piston moves up and down smoothly without friction, maintaining a dynamic equilibrium. The gas inside behaves ideally, providing consistent pressure across the piston face. This pressure is crucial for the subsequent calculations, including determining the piston鈥檚 mass and the gas column height.
Piston Force Calculation
To determine how much force the gas inside the tank exerts on the piston, we first need to understand the relationship between pressure and force. The fundamental formula is:
  • \( F = P \times A \)
Here, \( F \) represents the force, \( P \) is the pressure (converted to Pascals from atmosphere for consistency), and \( A \) is the area of the piston's circular face. The circular area can be found using:
  • \( A = \pi r^2 \)
With a radius of \(10.0 \text{ cm} = 0.10 \text{ m}\), we calculate the area and subsequently, the force applied by the gas on the piston.
Volume and Height Determination
In this scenario, we need to find out how tall the column of gas is that supports the piston. The Ideal Gas Law, expressed as:
  • \( PV = nRT \)
helps us calculate the total volume of gas within the tank, as it considers the number of moles, temperature, and gas constant. Once the volume \( V \) is determined using:
  • \( V = \frac{nRT}{P} \)
the height \( h \) of the gas column can be calculated using the volume-area-height relation:
  • \( h = \frac{V}{A} \)
This provides the necessary height of the gas column supporting the piston within the cylindrical tank.
Pressure Conversion and Calculation
Understanding pressure in different units is crucial. In these calculations, we begin with atmospheric pressure, typically used in ideal gas contexts, and convert it to Pascals, as pressure is often required in SI units for precise calculations:
  • \( 1.00 \text{ atm} = 101325 \text{ Pa} \)
This conversion is integral because it allows for dimensional consistency in equations. With the pressure correctly translated to Pascals, we can accurately proceed with calculations, ensuring the force on the piston and the final gas volume are precisely determined.

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Most popular questions from this chapter

For carbon dioxide gas \(\left(\mathrm{CO}_{2}\right),\) the constants in the van der Waals equation are \(a=0.364 \mathrm{J} \cdot \mathrm{m}^{3} / \mathrm{mol}^{2}\) and \(b=4.27 \times 10^{-5} \mathrm{m}^{3} / \mathrm{mol} .\) (a) If 1.00 \(\mathrm{mol}\) of \(\mathrm{CO}_{2} \mathrm{gas}\) at 350 \(\mathrm{K}\) is confined to a volume of 400 \(\mathrm{cm}^{3}\) , find the pressure of the gas using the ideal-gas equation and the van der Waals equation. (b) Which equation gives a lower pressure? Why? What is the percentage difference of the van der Waals equation result from the ideat-gas equation result? (c) The gas is kept at the same temperature as it expands to a volume of 4000 \(\mathrm{cm}^{3} .\) Repeat the calculations of parts (a) and (b). (d) Explain how your calculations show that the van der Waals equation is equivalent to the ideat-gas equation if \(n / V\) is small.

The surface of the sun has a temperature of about 5800 \(\mathrm{K}\) and consists largely of hydrogen atoms. (a) Find the rms speed of a hydrogen atom at this temperature. (The mass of a single hydrogen atom is 1.67 \(\times 10^{-27} \mathrm{kg} . )\) (b) The escape speed for a particle to leave the gravitational influence of the sun is given by \((2 G M / R)^{1 / 2}\) , where \(M\) is the sun's mass, \(R\) its radius, and \(G\) the gravitational constant (see Example 12.5 of Section \(12.3 ) .\) Use the data in Appendix \(\mathrm{F}\) to calculate this escape speed. (c) Can appreciable quantities of hydrogen escape from the sun? Can any hydrogen escape? Explain.

A large cylindrical tank contains 0.750 in \(^{3}\) of nitrogen gas at \(27^{\circ} \mathrm{C}\) and \(1.50 \times 10^{5} \mathrm{Pa}\) (absolute pressure). The tank has a tight-fitting piston that allows the volume to be changed. What will be the pressure if the volume is decreased to 0.480 \(\mathrm{m}^{3}\) and the temperature is increased to \(157^{\circ} \mathrm{C}\) ?

(a) Calculate the total rotational kinetic energy of the molecules in 1.00 mol of a diatomic gas at 300 \(\mathrm{K}\) . (b) Calculate the moment of inertia of an oxygen molecule \(\left(\mathrm{O}_{2}\right)\) for rotation about either the \(y\) - or \(z\) -axis shown in Fig. 18.18 . Treat the molecule as two massive points (representing the oxygen atoms) separated by a distance of \(1.21 \times 10^{-10} \mathrm{m}\) . The molar mass of oxygen atoms is \(16.0 \mathrm{g} / \mathrm{mol} .\) (c) Find the rms angular velocity of rotation of an oxygen molecule about either the \(y\) - or \(z\) -axis shown in Fig. 18.15. How does your answer compare to the angular velocity of a typical piece of rapidly rotating machinery \((10,000 \mathrm{rev} / \mathrm{min}) ?\)

A cylindrical tank has a tight-fitting piston that allows the volume of the tank to be changed. The tank originally contains 0.110 \(\mathrm{m}^{3}\) of air at pressure of 3.40 atm. The piston is slowly pulled out until the volume of the gas is increased to 0.390 \(\mathrm{m}^{3} .\) If the temperature remains constant, what is the final value of the pressure?
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