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Chapter 18: Problem 13

If a certain amount of ideal gas occupies a volume \(V\) at STP on earth, what would be its volume (in terms of \(V\)) on Venus, where the temperature is 1003\(^\circ\)C and the pressure is 92 atm?

Short Answer

Expert verified
The gas volume on Venus is approximately 0.0508 times its volume at STP on Earth.

Step by step solution

01

Understand the Ideal Gas Law

The ideal gas law \( PV = nRT \) relates the pressure \( P \), volume \( V \), temperature \( T \), and the number of moles \( n \) of a gas, where \( R \) is the ideal gas constant. At STP (standard temperature and pressure), the temperature is 273 K and the pressure is 1 atm.
02

Determine Known Values

Earth's STP conditions are: \( P_1 = 1 \) atm, \( T_1 = 273 \) K. Venus' conditions are: \( P_2 = 92 \) atm, \( T_2 = 1003 ^\circ C = 1003 + 273 = 1276 \) K. We are trying to find the volume \( V_2 \) on Venus, given that at STP \( V_1 = V \).
03

Apply the Combined Gas Law

The combined gas law \( \frac{P_1 V_1}{T_1} = \frac{P_2 V_2}{T_2} \) relates the volumes, pressures, and temperatures of a gas at two different states.
04

Solve for the Volume on Venus \( V_2 \)

Rearrange the combined gas law to solve for \( V_2 \): \[ V_2 = \frac{P_1 V_1 T_2}{P_2 T_1} \]Substitute the known values:\[ V_2 = \frac{1 \cdot V \cdot 1276}{92 \cdot 273} \]
05

Simplify the Expression

Calculate the expression:\[ V_2 = \frac{1276V}{92 \times 273} \] \[ \approx \frac{1276V}{25096} \approx 0.0508V \]
06

Conclude the Result

Thus, the volume of the gas on Venus is approximately 0.0508 of its original volume at STP on Earth.

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Key Concepts

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

Combined Gas Law
The Combined Gas Law is a powerful equation that helps us understand how gases behave under different conditions of pressure, temperature, and volume. It essentially combines three core gas laws - Boyle's Law, Charles's Law, and Gay-Lussac's Law into one. This law is expressed as:\[ \frac{P_1 V_1}{T_1} = \frac{P_2 V_2}{T_2} \]Here:
  • \(P_1\) and \(P_2\) are the initial and final pressures.
  • \(V_1\) and \(V_2\) are the initial and final volumes.
  • \(T_1\) and \(T_2\) are the initial and final temperatures in Kelvin.
This formula is extremely useful when you know the conditions of a gas at one state and you want to predict what happens if those conditions change. Always remember to convert temperatures to Kelvin, the absolute scale, to get accurate results.
Standard Temperature and Pressure (STP)
Standard Temperature and Pressure (STP) is a crucial reference point in chemistry and physics. It defines a set of conditions where:
  • The temperature is 273 Kelvin, equivalent to 0掳C.
  • The pressure is 1 atmosphere (atm).
STP is commonly used as a baseline to compare the properties of gases because it provides a known environment. When dealing with gas laws, such as the ideal gas law or combined gas law, using STP simplifies calculations because many values standardize at these conditions. For example, one mole of any ideal gas at STP occupies a volume of 22.4 liters.
Venus atmospheric conditions
Understanding Venus's atmospheric conditions is crucial for calculations involving gases on the planet. Venus presents extreme conditions when compared to Earth, making it a unique study topic. Let's consider a few key points about Venus:
  • The surface pressure is about 92 atm, which is much higher than Earth's 1 atm.
  • The average temperature on the surface can reach 1003掳C, or 1276 Kelvin when converted to the proper scale for gas laws.
These conditions result in significantly different behavior of gases compared to those on Earth. The extreme pressure and temperature lead to smaller volumes of gases as they are compressed under the heavy atmosphere of Venus. This understanding is pivotal when predicting the changes in a gas's volume or pressure when moved from Earth to Venus or vice versa.

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Most popular questions from this chapter

A physics lecture room at 1.00 atm and 27.0\(^\circ\)C has a volume of 216 m\(^3\). (a) Use the ideal-gas law to estimate the number of air molecules in the room. Assume that all of the air is N\(_2\). Calculate (b) the particle density-that is, the number of N\(_2 \) molecules per cubic centimeter-and (c) the mass of the air in the room.

(a) A deuteron, 21 \(H\), is the nucleus of a hydrogen isotope and consists of one proton and one neutron. The plasma of deuterons in a nuclear fusion reactor must be heated to about 300 million \(K\). What is the rms speed of the deuterons? Is this a significant fraction of the speed of light in vacuum (c = 3.0 \(\times\) 10\(^8\) m/s)? (b) What would the temperature of the plasma be if the deuterons had an rms speed equal to 0.10\(c\)?

A welder using a tank of volume 0.0750 m\(^3\) fills it with oxygen (molar mass 32.0 g/mol) at a gauge pressure of 3.00 \(\times\) 10\({^5}\) Pa and temperature of 37.0\(^\circ\)C. The tank has a small leak, and in time some of the oxygen leaks out. On a day when the temperature is 22.0\(^\circ\)C, the gauge pressure of the oxygen in the tank is 1.80 \(\times\) 10\({^5}\) Pa. Find (a) the initial mass of oxygen and (b) the mass of oxygen that has leaked out.

(a) Calculate the density of the atmosphere at the surface of Mars (where the pressure is 650 Pa and the temperature is typically 253 \(K\), with a \(CO_2\) atmosphere), Venus (with an average temperature of 730 \(K\) and pressure of 92 atm, with a \(CO_2\) atmosphere), and Saturn's moon Titan (where the pressure is 1.5 atm and the temperature is -178\(^\circ\)C, with a \(N_2\) atmosphere). (b) Compare each of these densities with that of the earth's atmosphere, which is 1.20 kg/m\(^3\). Consult Appendix D to determine molar masses.

You blow up a spherical balloon to a diameter of 50.0 cm until the absolute pressure inside is 1.25 atm and the temperature is 22.0\(^\circ\)C. Assume that all the gas is N\(_2\), of molar mass 28.0 g/mol. (a) Find the mass of a single N\(_2\) molecule. (b) How much translational kinetic energy does an average N\(_2\) molecule have? (c) How many N\(_2\) molecules are in this balloon? (d) What is the \(total\) translational kinetic energy of all the molecules in the balloon?
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