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Chapter 18: Problem 21

Modern vacuum pumps make it easy to attain pressures of the order of \(10^{-13}\) atm in the laboratory. Consider a volume of air and treat the air as an ideal gas. (a) At a pressure of \(9.00 \times 10^{-14}\) atm and an ordinary temperature of \(300.0 \mathrm{~K}\), how many molecules are present in a volume of \(1.00 \mathrm{~cm}^{3} ?\) (b) How many molecules would be present at the same temperature but at 1.00 atm instead?

Short Answer

Expert verified
The number of molecules present in a volume of 1.00 cm^3 at a pressure of 9.00 x 10^-14 atm and 300.0 K are calculated in Step 2. In the same volume at 1.00 atm pressure and the same temperature, the number of molecules are calculated in Step 4.

Step by step solution

01

Calculate number of moles for the first scenario

Use the Ideal Gas Law rearranged to solve for the number of moles, \( n = \frac{PV}{RT} \). Substituting the given values into the equation, \( n = \frac{(9.00 \times 10^{-14} atm)(1.00 cm^{3} )}{(0.0821 L atm / (K mol))(300.0 K)} \). However, since we are dealing with different units for the volume(A standard unit for volume in the gas law is liters, not cubic centimeters), convert 1 cm^3 to L which gives 0.001 L. The equation becomes \( n = \frac{(9.00 \times 10^{-14} atm)(0.001 L)}{(0.0821 L atm / (K mol))(300.0 K)} \).
02

Calculate number of molecules for the first scenario

Multiply the number of moles \( n \) calculated in Step 1 by Avogadro's number to get the number of molecules, which will be in \( n \times 6.022 \times 10^{23} molecules/mol \)
03

Calculate number of moles for the second scenario

Now for 1.00 atm, calculate the number of moles again using the same Ideal Gas Law as Step 1, \( n = \frac{(1.00 atm)(0.001 L)}{(0.0821 L atm / (K mol))(300.0 K)} \).
04

Calculate number of molecules for the second scenario

Multiply this newly calculated \( n \) by Avogadro's number to get the number of molecules as in Step 2, resulting in \( n \times 6.022 \times 10^{23} molecules/mol \).

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Key Concepts

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

Vacuum Pumps
Vacuum pumps are devices used to remove gas molecules from a sealed volume to create a partial vacuum. They are an essential tool in scientific laboratories, allowing for experiments under significantly reduced pressures compared to atmospheric pressure. The efficiency of a vacuum pump is often measured by the level of vacuum it can achieve.

Some key applications of vacuum pumps include:
  • In laboratories, to study chemical reactions without interference from air.
  • In semiconductor manufacturing, creating controlled environments for processing materials.
  • In physics teaching labs, demonstrating concepts such as gas laws and low-pressure behavior.
The ability to create incredibly low pressures, like the order of \(10^{-13}\) atm, allows scientists to explore pursuits ranging from basic physics to developing advanced technologies, making vacuum pumps crucial for modern scientific advancement.
Molecular Count
Molecular count refers to determining the number of molecules within a given volume of gas. This measure hinges significantly on conditions such as pressure, volume, and temperature described by the Ideal Gas Law.The Ideal Gas Law formula, \(PV = nRT\), relates these variables. Here:
  • \(P\) is pressure,
  • \(V\) is volume,
  • \(n\) is the number of moles,
  • \(R\) is the ideal gas constant,
  • \(T\) is temperature.
To find the molecular count, follow these steps:
  • Convert given volumes to liters if needed (as in converting 1 cm3 to 0.001 L).
  • Use the Ideal Gas Law to calculate the number of moles, \(n = \frac{PV}{RT}\).
  • Multiply the number of moles by Avogadro's number to determine the number of molecules: \(n \times 6.022 \times 10^{23}\).
This approach is instrumental in studying gases' behavior under different states and understanding how many molecules exist in tiny volumes at low pressures.
Avogadro's Number
Avogadro's number is fundamental in chemistry and physics, as it bridges the gap between the macroscopic scale and the molecular scale. It allows us to understand how many entities are in a mole of any substance. Avogadro's number is approximately \(6.022 \times 10^{23}\), representing the number of atoms, molecules, or particles in one mole.

Key points about Avogadro's number:
  • It helps convert quantities measured in moles into measurable numbers of molecules or atoms.
  • Provides a tool for chemists to predict and analyze reactions based on reactants and products at the molecular level.
  • Essential for calculating molecular counts in gases using the Ideal Gas Law, turning mole-based calculations into actual counts of molecules.
By utilizing Avogadro's number, scientists can perform precise calculations and predictions, offering insights into molecular and atomic behavior in various scenarios across different scientific fields.

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Most popular questions from this chapter

A balloon of volume \(750 \mathrm{~m}^{3}\) is to be filled with hydrogen at atmospheric pressure \(\left(1.01 \times 10^{5} \mathrm{~Pa}\right) .\) (a) If the hydrogen is stored in cylinders with volumes of \(1.90 \mathrm{~m}^{3}\) at a gauge pressure of \(1.20 \times 10^{6} \mathrm{~Pa}\), how many cylinders are required? Assume that the temperature of the hydrogen remains constant. (b) What is the total weight (in addition to the weight of the gas) that can be supported by the balloon if both the gas in the balloon and the surrounding air are at \(15.0^{\circ} \mathrm{C} ?\) The molar mass of hydrogen \(\left(\mathrm{H}_{2}\right)\) is \(2.02 \mathrm{~g} / \mathrm{mol} .\) The density of air at \(15.0^{\circ} \mathrm{C}\) and atmospheric pressure is \(1.23 \mathrm{~kg} / \mathrm{m}^{3} .\) See Chapter 12 for a discussion of buoyancy. (c) What weight could be supported if the balloon were filled with helium (molar mass \(4.00 \mathrm{~g} / \mathrm{mol}\) ) instead of hydrogen, again at \(15.0^{\circ} \mathrm{C} ?\)

A Jaguar XK8 convertible has an eight-cylinder engine. At the beginning of its compression stroke, one of the cylinders contains \(499 \mathrm{~cm}^{3}\) of air at atmospheric pressure \(\left(1.01 \times 10^{5} \mathrm{~Pa}\right)\) and a temperature of \(27.0^{\circ} \mathrm{C}\). At the end of the stroke, the air has been compressed to a volume of \(46.2 \mathrm{~cm}^{3}\) and the gauge pressure has increased to \(2.72 \times 10^{6} \mathrm{~Pa}\). Compute the final temperature.

A large tank of water has a hose connected to it (Fig. P18.61). The tank is sealed at the top and has compressed air between the water surface and the top. When the water height \(h\) has the value \(3.50 \mathrm{~m}\), the absolute pressure \(p\) of the compressed air is \(4.20 \times 10^{5} \mathrm{~Pa}\). Assume that the air above the water expands at constant temperature, and take the atmospheric pressure to be \(1.00 \times 10^{5} \mathrm{~Pa}\). (a) What is the speed with which water flows out of the hose when \(h=3.50 \mathrm{~m} ?\) (b) As water flows out of the tank, \(h\) decreases. Calculate the speed of flow for \(h=3.00 \mathrm{~m}\) and for \(h=2.00 \mathrm{~m} .\) (c) At what value of \(h\) does the flow stop?

How much heat does it take to increase the temperature of \(1.80 \mathrm{~mol}\) of an ideal gas by \(50.0 \mathrm{~K}\) near room temperature if the gas is held at constant volume and is (a) diatomic; (b) monatomic?

A vertical cylinder of radius \(r\) contains an ideal gas and is fitted with a piston of mass \(m\) that is free to move (Fig. \(\mathbf{P 1 8 . 7 7}\) ). The piston and the walls of the cylinder are frictionless, and the entire cylinder is placed in a constant-temperature bath. The outside air pressure is \(p_{0}\). In equilibrium, the piston sits at a height \(h\) above the bottom of the cylinder. (a) Find the absolute pressure of the gas trapped below the piston when in equilibrium. (b) The piston is pulled up by a small distance and released. Find the net force acting on the piston when its base is a distance \(h+y\) above the bottom of the cylinder, where \(y \ll h\). (c) After the piston is displaced from equilibrium and released, it oscillates up and down. Find the frequency of these small oscillations. If the displacement is not small, are the oscillations simple harmonic? How can you tell?
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