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Chapter 14: Problem 21

On the sunlit surface of Venus, the atmospheric pressure is \(9.0 \times 10^{6} \mathrm{Pa},\) and the temperature is 740 \(\mathrm{K}\) . On the earth's surface the atmospheric pressure is \(1.0 \times 10^{5} \mathrm{Pa}\) , while the surface temperature can reach 320 \(\mathrm{K}\) . These data imply that Venus has a "thicker" atmosphere at its surface than does the earth, which means that the number of molecules per unit volume \((N / V)\) is greater on the surface of Venus than on the earth. Find the ratio \((N / V)_{\mathrm{Venus}} /(N / V)_{\mathrm{Earth}}\)

Short Answer

Expert verified
The ratio \( (N/V)_{\text{Venus}} / (N/V)_{\text{Earth}} \) is approximately 39.0.

Step by step solution

01

Understanding the Ideal Gas Law

To find the ratio \( (N/V)_{\text{Venus}} / (N/V)_{\text{Earth}} \), we will use the ideal gas law, which is normally given by \( PV = NkT \), where \( P \) is pressure, \( V \) is volume, \( N \) is the number of molecules, \( k \) is the Boltzmann constant, and \( T \) is temperature. Rearranging this formula to find \( N/V \) gives us \( N/V = P/(kT) \).
02

Calculating \( (N/V)_{\text{Venus}} \)

For Venus, we have \( P_{\text{Venus}} = 9.0 \times 10^{6} \mathrm{Pa} \) and \( T_{\text{Venus}} = 740 \mathrm{K} \). Using the rearranged ideal gas formula, the number density \( N/V \) is \( \frac{P_{\text{Venus}}}{kT_{\text{Venus}}} \).
03

Calculating \( (N/V)_{\text{Earth}} \)

For Earth, the pressure \( P_{\text{Earth}} = 1.0 \times 10^{5} \mathrm{Pa} \) and the temperature \( T_{\text{Earth}} = 320 \mathrm{K} \). Again, use the formula: \( \frac{P_{\text{Earth}}}{kT_{\text{Earth}}} \).
04

Finding the Ratio

To find the ratio \( (N/V)_{\text{Venus}} / (N/V)_{\text{Earth}} \), divide \( (N/V)_{\text{Venus}} \) by \( (N/V)_{\text{Earth}} \). The \( k \) in the numerator and denominator cancels out, so the ratio is simply \( \frac{P_{\text{Venus}}/T_{\text{Venus}}}{P_{\text{Earth}}/T_{\text{Earth}}} \).
05

Calculating the Numerical Ratio

Plug in the given values into the ratio: \( \frac{9.0 \times 10^{6} / 740}{1.0 \times 10^{5} / 320} = \frac{9.0 \times 10^{6} \times 320}{1.0 \times 10^{5} \times 740} \). Simplify this expression to find the final ratio.

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Key Concepts

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

Atmospheric Pressure
Atmospheric pressure is a measure of the force exerted by the atmosphere on a surface. This force is due to the weight of the air molecules above that surface. On Earth, atmospheric pressure at sea level is approximately \(1.0 \times 10^{5}\, \text{Pa}\) (pascals). However, on Venus, it rockets to a much higher value of \(9.0 \times 10^{6}\, \text{Pa}\).

Why such a big difference? Here are the factors:
  • Thickness of Atmosphere: Venus has a denser atmosphere composed largely of carbon dioxide, while Earth's is mostly nitrogen and oxygen.
  • Gravitational Pull: Venus' gravitational pull contributes to retaining a thick atmospheric layer.
  • Distance from Sun: This affects temperature, which in turn impacts gas particle activity contributing to pressure levels.
Understanding these fundamental pressure differences is essential in explaining atmospheric behaviors on these two planets.
Temperature on Venus
Temperature plays a crucial role in planetary atmospheres and their dynamics. Venus is known for its extreme temperatures, reaching up to \(740\, \text{K}\) on its sunlit side. This high temperature is primarily due to two factors:
  • Greenhouse Effect: Thick clouds rich in COevolve, retain the sun's heat, resulting in scorching surface temperatures.
  • Proximity to the Sun: Being closer to the Sun compared to Earth adds to the heat, although atmospheric composition is the main reason for such high temperatures.
On Earth, temperatures can reach up to \(320\, \text{K}\), significantly lower compared to Venus. The planetary temperature impacts how fast gas molecules move and the type of atmospheric pressure generated, helping to shape weather patterns and climate on each planet.
Molecular Density Ratio
The concept of molecular density ratio, \((N/V)\), is derived from the ideal gas law. It refers to the number of molecules present per unit volume in an atmosphere. To understand why Venus has a greater molecular density, we work with the formula:

\[N/V = \frac{P}{kT}\]

Here, \(P\) stands for pressure, \(T\) for temperature, and \(k\) represents the Boltzmann constant. Both Venus and Earth are analyzed under this framework:
  • Venus: High pressure and relatively lower temperature increases \(N/V\), indicating a denser atmosphere.
  • Earth: Lower pressure with a higher temperature compared to Venus results in a lower \(N/V\).
Thus, when calculating the ratio \((N/V)_{\text{Venus}} / (N/V)_{\text{Earth}}\), factors like high pressure and high greenhouse effect of Venus contribute to a significantly higher molecular density ratio than that of Earth.

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Most popular questions from this chapter

A gas fills the right portion of a horizontal cylinder whose radius is 5.00 \(\mathrm{cm} .\) The initial pressure of the gas is \(1.01 \times 10^{5}\) Pa. A frictionless movable piston separates the gas from the left portion of the cylinder, which is evacuated and contains an ideal spring, as the drawing shows. The piston is initially held in place by a pin. The spring is initially unstrained, and the length of the gas-filled portion is 20.0 \(\mathrm{cm}\) . When the pin is removed and the \(\mathrm{gas}\) is allowed to expand, the length of the gas-filled chamber doubles. The initial and final temperatures are equal. Determine the spring constant of the spring.

Manufacturers of headache remedies routinely claim that their own brands are more potent pain relievers than the competing brands. Their way of making the comparison is to compare the number of molecules in the standard dosage. Tylenoluses 325 \(\mathrm{mg}\) of acetaminophen \(\left(\mathrm{C}_{8} \mathrm{H}_{9} \mathrm{NO}_{2}\right)\) as the standard dose, whereas Advil uses \(2.00 \times 10^{2} \mathrm{mg}\) of ibuprofen \(\left(\mathrm{C}_{13} \mathrm{H}_{18} \mathrm{O}_{2}\right) .\) Find the number of molecules of pain reliever in the standard doses of \((\mathrm{a})\) Tylenol and \(\quad\) (b) Advil.

The chlorophyll-a molecule \(\left(\mathrm{C}_{55} \mathrm{H}_{72} \mathrm{MgN}_{4} \mathrm{O}_{5}\right)\) is important in photosynthesis. (a) Determine its molecular mass (in atomic mass units). (b) What is the mass (in grams) of 3.00 moles of chlorophyll-a molecules?

When a gas is diffusing through air in a diffusion channel, the diffusion rate is the number of gas atoms per second diffusing from one end of the channel to the other end. The faster the atoms move, the greater is the diffusion rate, so the diffusion rate is proportional to the rms speed of the atoms. The atomic mass of ideal gas A is 1.0 u, and that of ideal gas B is 2.0 u. For diffusion through the same channel under the same conditions, find the ratio of the diffusion rate of gas A to the diffusion rate of gas B.

When you push down on the handle of a bicycle pump, a piston in the pump cylinder compresses the air inside the cylinder. When the pressure in the cylinder is greater than the pressure inside the inner tube to which the pump is attached, air begins to flow from the pump to the inner tube. As a biker slowly begins to push down the handle of a bicycle pump, the pressure inside the cylinder is \(1.0 \times 10^{5} \mathrm{Pa},\) and the piston in the pump is 0.55 \(\mathrm{m}\) above the bottom of the cylinder. The pressure inside the inner tube is \(2.4 \times 10^{5}\) Pa. How far down must the biker push the handle before air begins to flow from the pump to the inner tube? Ignore the air in the hose connecting the pump to the inner tube, and assume that the temperature of the air in the pump cylinder does not change.
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